Free Geometry Problems and Questions writh Solutions

Free geometry tutorials on topics such as perpendicular bisector, central and inscribed angles, circumcircles, sine law and triangle properties to solve triangle problems. Also geometry problems with detailed solutions on triangles, polygons, parallelograms, trapezoids, pyramids and cones are included. Polar coordinates equations, conversion and graphing are also included. More challenging geometry problems are also included.

Geometry Problems

  • Triangle Problems . Triangle problems with detailed solutions.
  • Congruent Triangles Examples and Problems with Solutions .
  • Similar Triangles Examples and Problems with Solutions . Definition and theorems on similar triangles including examples and problems with detailed solutions.
  • Equilateral Triangles Problems with Solutions .
  • Isosceles Triangles Problems with Solutions .
  • Area and Perimeter of Right Triangles Problems With Solution .
  • Cosine Law Problems . The cosine law is used to solve word problems.
  • Sine Law Problems . The sine law is used to solve word problems.
  • Triangle Inscribed in a Circle - Problem With Solution . Inscribed right triangle problem with detailed solution.
  • Circle Tangent to Right Triangle - Problem With Solution . Solve a right triangle whose sides are all tangent to a circle. Both the problem and its detailed solution are presented.
  • Overlapping Circles Problem . Find the overlapping area of two circles: problem with detailed solutions.
  • Sectors and Circles Problems . Problems, with detailed solutions, related to sectors and circles.
  • Two Squares and a Circle - Problem With Solution . A problem, with a detailed solution, on a circle inscribed in one square and circumscribed to another, is presented.
  • Two Circles and a Square - Problem With Solution . A problem, with a detailed solution, on a square inscribed in one circle and circumscribed to another is presented.

Quadrilaterals

  • Rectangle Problems . Rectangle problems on area, dimensions, perimeter and diagonal with detailed solutions.
  • Geometry Problems on Squares . Square problems on area, diagonal and perimeter, with detailed solutions.
  • Parallelogram Problems . Word problems related to parallelograms are presented along with detailed solutions.
  • Trapezoid Problems . Trapezoid problems are presented along with detailed solutions.
  • Solve a Trapezoid Given its Bases and Legs .
  • Rhombus Problems . Rhombus definition and properties are presented along with problems with detailed solutions.
  • Polygons Problems . Problems related to regular polygons.
  • Area Of Octagon - Problem With Solution . Find the length of one side, the perimeter and area of a regular octagon given the distance between two opposite sides (span).
  • Angles in Parallel Lines and Transversals Problems . Problems related to parallel lines and alternate and corresponding angles.
  • 3D Shapes Volume Problems . 3D shapes, such as prisms, volume problems with detailed solutions.
  • Compare Volumes of 3D shapes . A problem to compare the volumes of a cone, a cylinder and a hemisphere.
  • How to construct a frustum? . If you cut off the top part of a cone with a plane perpendicular to the height of the cone, you obtain a conical frustum. How to construct a frustum given the radius of the base, the radius of the top and the height?
  • Cone Problems . Problems related to the surface area and volume of a cone with detailed solutions are presented.
  • Pyramid Problems . Pyramid problems related to surface area and volume with detailed solutions.
  • Intercept Theorem and Problems with Solutions .

Geometry Tutorials

  • Parts of a Circle .
  • Tangents to a Circle with Questions and Solutions .
  • Intersecting Secant and Tangent Theorem Questions with Solutions .
  • Inscribed and Central Angles in Circles . Definitions and theorems related to inscribed and central angles in circles are discussed using examples and problems.
  • Intersecting Chords Theorem Questions with Solutions.
  • Intersecting Secant Theorem Questions with Solutions.
  • Semicircle Thales Theorem with questions and solutions
  • Central and Inscribed Angles - Interactive applet . The properties of central and inscribed angles intercepting a common arc in a circle are explored using an interactive geometry applet.
  • Triangles . Definitions and properties of triangles in geometry.
  • Area of Triangles Problems with Solutions . Use different formulas of the area of a triangle to calculate the areas of triangles and shapes.
  • Simple Proofs of Pythagorean Theorem and Problems with Solutions .
  • Altitudes, Medians and Angle Bisectors of a Triangle .
  • Triangles, Bisectors and Circumcircles - interactive applet . The properties of perpendicular bisectors in triangles and circumcircles are explored interactively using a geometry Java applet.
  • Quadrilaterals , properties and formulas.
  • Kite Questions with Solutions .
  • Angles in Geometry . Definitions and properties of angles in geometry including questions with solutions.
  • Angles in Parallel Lines and Transversals . This tutorial is about the corresponding, interior and exterior angles formed when a transversal line intersects two parallel lines.
  • Latitude and Longitude Coordinate System .
  • Find the GPS Latitude and Longitude Using Google Map .
  • Regular Polygons . Tutorial to develop useful formulas for area of regular polygons.

Other Geometry Topics

  • Perpendicular Bisector Problems with Solutions .
  • Table of Formulas for Geometry . A table of formulas for geometry, related to area and perimeter of triangles, rectangles, circles, sectors, and volume of sphere, cone, cylinder are presented.

Challenge Geometry Problems

  • Two Tangent Circles and a Square - Problem With Solution . You are given the perimeter of a small circle to find the radius of a larger circle inscribed within a square.
  • Kite Within a Square - Problem With Solution . A problem on finding the sine of the angle of a kite within a square.
  • Solve Triangle Given Its Perimeter, Altitute and Angle - Problem With Solution .
  • Solve Right Triangle Given Perimeter and Altitude - Problem With Solution .
  • Triangle and Tangent Circle - Problem With Solution . A problem, on a triangle tangent at two points to a circle, is presented along with detailed solution.
  • Three Tangent Circles - Problem With Solution . A problem, on three tangent circles, is presented along with solution.
  • Equilateral Triangle Within a Square - Problem With Solution . A problem on the proof of an equilateral triangle within a square is presented along with detailed solution.
  • Square Inscribed in Right Triangle - Problem With Solution . Find the side of a square inscribed in a right triangle given the sides of the triangle.

Polar Coordinates

  • Plot Points in Polar Coordinates . An interactive tutorial on how to plot points given by their polar coordinates.
  • Graphing Polar Equations . This is tutorial on graphing polar equations by hand, or sketching, to help you gain deep understanding of these equations. Several examples with detailed solutions are presented.
  • Convert Polar to Rectangular Coordinates and Vice Versa . Problems, with detailed solutions, where polar coordinates are converted into rectangular coordinates and vice versa are presented.
  • Convert Equation from Rectangular to Polar Form . Problems were equations in rectangular form are converted to polar form, using the relationship between polar and rectangular coordinates, are presented along with detailed solutions.
  • Convert Equation from Polar to Rectangular Form . Equations in polar form are converted into rectangular form, using the relationship between polar and rectangular coordinates. Problems with detailed solutions are presented.

Geometric Transformations

  • Reflection Across a Line . The properties of reflection of shapes across a line are explored using a geometry applet.
  • Rotation of Geometric Shapes . The rotations of 2-D shapes are explored.

Geometric Calculators and Worksheets

  • Online Geometry Calculators and Solvers : Several calculators to help in the calculations and solutions of geometry problems.
  • Free Geometry Worksheets to Download

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Using Geometric Concepts and Properties to Solve Problems

Introduction.

Often, you will be asked to solve problems involving geometric relationships or other shapes. For real-world problems, those geometric relationships mostly involve measurable attributes, such as length, area, or volume.

Sometimes, those problems will involve the perimeter or circumference, or the area of a 2-dimensional figure.

green elliptical running track

For example, what is the distance around the track that is shown?  Or, what is the area of the portion of the field that is covered with grass?

You may also see problems that involve the volume or surface area of a 3-dimensional figure.  For example, what is the area of the roof of the building that is shown?

building composed of a rectangular prism with a half-cylinder on top

Another common type of geometric problem involves using proportional reasoning.

how to solve geometry related problems

For example, an artist created a painting that needs to be reduced proportionally for the flyer advertising an art gallery opening. If the dimensions of the painting are reduced by a factor of 40%, what will be the dimensions of the image on the flyer?

In this resource, you will investigate ways to apply a problem-solving model to determine the solutions for geometric problems like these.

A basic problem solving model contains the following four steps:

Solving Problems Using Perimeter and Circumference

You may recall that the perimeter of an object is the distance around the edge of the object. If the object contains circles, then you may need to think about the circumference of a circle, which is the perimeter of the circle.

A tire on a passenger car has a diameter of 18 inches. When the tire has rotated 5 times, how far will the car have traveled?

Image of car and its tire labeled 18 inches for diameter

Step 1 : Read, understand, and interpret the problem.

  • What information is presented?
  • What is the problem asking me to find?
  • What information may be extra information that I do not need?

Step 2 : Make a plan.

  • Draw a picture.
  • Use a formula: Which formula do I need to use? (Hint: Look at your Mathematics Reference Materials)

Step 3 : Implement your plan.

  • What formulas do I need?
  • What information can I interpret from the diagram, table, or other given information?
  • Solve the problem.

Step 4 : Evaluate your answer.

  • Does my answer make sense?
  • Did I answer the question that was asked?
  • Are my units correct?

A cylindrical barrel with a diameter of 20 inches is used to hold fuel for a barbecue cook off. The chef rolls the barrel so that it completes 7 rotations. How many feet did the chef roll the barrel?

Image of barrel with diameter labeled 20 inches

Solving Problems Using Area and Surface Area

You may also encounter real-world geometric problems that ask you to find the area of 2-dimensional figures or the surface area of 3-dimensional figures. The key to solving these problems is to look for ways to break the region into smaller figures of which you know how to find the area.

Mr. Elder wants to cover a wall in his kitchen with wallpaper. The wall is shown in the figure below.

hexagonal wall with dimensions labeled

If wallpaper costs $1.75 per square foot, how much will Mr. Elder spend on wallpaper to completely cover this wall, excluding sales tax?

To solve this problem, let's use the 4-step problem solving model.

Mrs. Nguyen wants to apply fertilizer to her front lawn. A bag of fertilizer that covers 1,000 square feet costs $18. How many bags of fertilizer will Mrs. Nguyen need to purchase?

hexagonal yard with dimensions labeled

Surface Area Problem

After a storm, the Serafina family needs to have their roof replaced. Their house is in the shape of a pentagonal prism with the dimensions shown in the diagram.

pentagonal prism shaped house with roof shaded

To match their new roof, Mrs. Serafina decided to have both pentagonal sides of their house covered in aluminum siding. Their house is in the shape of a pentagonal prism with the dimensions shown in the diagram.

pentagonal prism shaped house with roof shaded

A contractor gave Mrs. Serafina an estimate based on a cost of $3.10 per square foot to complete the aluminum siding. How much will it cost the Serafina family to have the aluminum siding installed?

Solving Problems Using Proportionality

Proportional relationships are another important part of geometric problem solving.

A woodblock painting has dimensions of 60 centimeters by 79.5 centimeters. In order to fit on a flyer advertising the opening of a new art show, the image must be reduced by a scale factor of  1/25.

W hat will be the final dimensions of the image on the flyer?

Image of the sun over a Japanese temple

Measuring Problem

For summer vacation, Jennifer and her family drove from their home in Inlandton to Beachville. Their car can drive 20 miles on one gallon of gasoline. Use the ruler to measure the distance that they drove to the nearest  1/4  inch, and then calculate the number of gallons of gasoline their car will use at this rate to drive from Inlandton to Beachville.

Practice #1

A blueprint for a rectangular tool shed has dimensions shown in the diagram below.

blueprint showing that the length is 4.5 centimeters and the width is 3.5 centimeters, and a scale of 1 centimeter = 2 feet

Todd is using this blueprint to build a tool shed, and he wants to surround the base of the tool shed with landscaping timbers as a border. How many feet of landscaping timbers will Todd need?

Practice #2

A scale model of a locomotive is shown. Use the ruler to measure the dimensions of the model to the nearest 1/4  inch, and then calculate the actual dimensions of the locomotive.

Scale : 1 inch = 5 feet

Solving geometric problems, such as those found in art and architecture, is an important skill. As with any mathematical problem, you can use the 4-step problem solving model to help you think through the important parts of the problem and be sure that you don't miss key information.

There are a lot of different applications of geometry to real-world problem solving. Some of the more common applications include the following:

octagonal cup

What is the perimeter of the base of the cup, if the cup is in the shape of an octagonal prism?

aerial photo showing crop circles

The JP Morgan Chase Bank Tower in downtown Houston, Texas, is one of the tallest buildings west of the Mississippi River. It is in the shape of a pentagonal prism. If 40% of each face is covered with glass windows, what is the amount of surface area covered with glass?

image of Van Gogh's Starry Night

The dimensions of Vincent van Gogh's Starry Night are 29 inches by 36 1 4 36\frac{1}{4} inches. If a print reduces these dimensions by a scale factor of 30%, what will be the dimensions of the print?

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o Identify some critical steps of the process for solving practical geometry problems

o Apply geometry problem-solving techniques to practical situations

Geometry has a variety of real-life applications in everyday situations. In this article, we will learn to apply geometric principles and techniques to solve problems. The key to solving practical geometry problems is translation of the real-life situation into figures, measurements, and other information necessary to represent the situation conceptually. For instance, you already know how to calculate the area of a composite figure; if you were asked to determine how much floor space is available in a certain building with a composite shape, you would simply need to apply the same principles as you would use for calculating the area of a composite figure. Some measurements of the building might, of course, be required, but the same problem-solving techniques apply.

It behooves us to present a basic approach to solving practical geometry problems. This approach is similar to that for solving almost a word problem, but is geared slightly more toward the characteristics of geometry problems in particular.

1. Determine what you need to calculate to solve the problem. In some cases, you may need a length; in others, an area or angle measure. If you are conscious throughout the process of what you need to determine, you can save yourself a significant amount of time.

2. Draw a diagram. Sometimes a straightedge, compass, protractor, or some combination of these tools can be helpful. Even if you only use a rough sketch, however, making a visual representation of the problem can help you organize your thoughts and keep track of important information such as the relationship of line segments and angles as well as the measures thereof.

3. Record all appropriate measurements. If you are calculating an area, for instance, you may need to take measurements of certain lengths (alternatively, these may be provided to you). In either case, record them and mark them in some manner on your diagram.

4. Pay attention to units. Using units of square meters for a length or angle measure can be an embarrassing mistake! Keep careful track of the units you are using throughout the problem. If no units are given, simply use the generic term "units" in place of inches or meters, for example.

5. Divide the figure, if necessary, into manageable portions. If your diagram is a composite figure, it may help to divide the figure into bite-sized portions that you can handle.

6. Identify any appropriate geometric relationships. This step can greatly simplify the problem. Perhaps you can show two triangles to be congruent or similar, or perhaps you can identify congruent segments or angles. Use this step to fill in as much missing information in your diagram as you can.

7. Do the math. At this point, you need to apply what you've learned to analyze the figure and other data to solve the problem. You may, for example, need to apply the Pythagorean theorem, or you may need to calculate the perimeter of a figure. Whatever the details of the problem, you will need to apply your skills in geometry in an appropriate manner.

8. Check your results. Take a look at your answer in the context of your diagram-does your answer make sense? A result of millions of square meters for the area of a figure with dimensions in the range of a few meters should tell you that you've made an error at some point in your analysis.

Not every step of the approach outlined above will be needed in every problem. You must use your best judgment in determining what is necessary to solve the problem in a satisfactory and time-efficient manner. Also, you may not always think to use the exact progression of steps above; the outline is simply a way to describe a systematic approach to problem solving. The remainder of this article provides you the opportunity to test your geometry skills by way of several practice problems. Obviously, these problems do not require you to go out and make any measurements of lengths or angles, but keep in mind that problems you encounter in everyday life may require you to do so!

Practice Problem : The floor plan of a house is shown below. Determine the area covered by the house.

Solution : Let's first divide the diagram of the house into two rectangles and a trapezoid, since we can calculate the area of each of these figures. Using the properties of each figure, we can also fill in some of the unknown information.

Now, the area of the larger rectangle is the product of 40 feet and 20 feet, or 800 square feet. The area of the smaller rectangle is 25 feet times 6 feet, or 150 square feet. The area of the trapezoid is the following:

The height ( h ) is 6 feet, and the two bases ( b 1 and b 2 ) are 8 and 11 feet.

Adding all three areas gives us a total area of the house of 1,007 square feet.

Practice Problem : A hiker is walking up a steep hill. The slope of the hill between two trees is constant, and the base of one tree is 100 meters higher than the other. If the horizontal distance between the trees is 400 meters, how far must the hiker walk to get from one tree to the next?

Solution : Because this problem may be difficult to envision, a diagram is extremely helpful. Notice that the base of the trees differ in height by 100 meters--this is our vertical distance for the walk. The horizontal distance is 400 meters.

Note that we have shown the right angle because horizontal and vertical segments are perpendicular. We can now use the Pythagorean theorem to calculate the distance d the hiker must walk.

Thus, the hiker must walk about 412 meters. Note that although the hiker makes a significant (100 meter) change in elevation over this walk, the difference between the actual distance he walks and the horizontal distance is small--only about 12 meters.

Practice Problem : A homeowner has a rectangular fenced-in yard, and he wants to put mulch on his triangular gardens, as shown below. The inside border of each garden always meets the fence at the same angle. If a bag of mulch covers about 50 square feet, how many bags of mulch should the homeowner buy to cover his gardens?

Solution : We are told in the problem that the inside border of each garden meets the fence at the same angle in every case; thus, we can conclude (as shown below) that the triangles are all isosceles (and that the triangles with the same side lengths are congruent by the ASA condition). We can thus mark each side with an unknown variable x or y .

Recall that the fenced-in area is rectangular; thus the angle in each corner is 90°. We can then solve for x and y using the Pythagorean theorem. Notice first, however, that x and y are the height and base of their respective triangles.

Because the gardens include two of each triangle shape, the total garden area is simply the sum of x 2 and y 2 . (If you do not follow this point, simply use the triangle area formula in each case--you will get the same result.)

Thus, the homeowner needs six bags of mulch (for a total of 300 square feet) to cover his gardens. (Of course, we are assuming here that he must buy a whole number of bags.)

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Geometry Formulas

Geometry formulas are used for finding dimensions, perimeter, area, surface area, volume, etc. of the geometric shapes. Geometry is a part of mathematics that deals with the relationships of points, lines, angles, surfaces, solids measurement, and properties. There are two types of geometry: 2D or plane geometry and 3D or solid geometry.

The 2D shapes are flat shapes that have only two dimensions, length, and width as in squares, circles, and triangles, etc. 3D objects are solid objects, that have three dimensions, length, width, and height or depth, as in a cube, cuboid, sphere, cylinder, cone. Let us learn all geometry formulas along with a few solved examples in the upcoming sections.

What are Geometry Formulas?

The formulas used for finding dimensions, perimeter , area , surface area , volume , etc. of 2D and 3D geometric shapes are known as geometry formulas. 2D shapes consist of flat shapes like squares , circles , and triangles , etc., and cube , cuboid , sphere , cylinder , cone , etc are some examples of 3D shapes . The basic geometry formulas are given as follows:

basic Geometry formulas

Basic Geometry Formulas

Let us see the list of all Basic Geometry Formulas here.

2D Geometry Formulas

Here is the list of various 2d geometry formulas according to the geometric shape. It also includes a few formulas where the mathematical constant π(pi) is used.

  • Perimeter of a Square = 4(Side)
  • Perimeter of a Rectangle = 2(Length + Breadth)
  • Area of a Square = Side 2
  • Area of a Rectangle = Length × Breadth
  • Area of a Triangle = ½ × base × height
  • Area of a Trapezoid = ½ × (base 1 + base 2 ) × height
  • Area of a Circle = A = π×r 2

Circumference of a Circle = 2πr

3D Geometry Formulas

The basic 3D geometry formulas are given as follows. It should be noted that the following formulas have used the mathematical constant π(pi)

  • The curved surface area of a Cylinder = 2πrh
  • Total surface area of a Cylinder = 2πr(r + h)
  • Volume of a Cylinder = V = πr 2 h
  • The curved surface area of a cone = πrl
  • Total surface area of a cone = πr(r+l) = πr[r+√(h 2 +r 2 )]
  • Volume of a Cone = V = ⅓×πr 2 h
  • Surface Area of a Sphere = S = 4πr 2
  • Volume of a Sphere = V = 4/3×πr 3
  • r = Radius ;
  • h = Height. and,
  • l = Slant height

The formula table depicts the 2D geometry formulas and 3D geometry formulas.

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Let us have a look at solved examples to understand the basic geometry formulas.

Solved Examples Using Geometry Formulas

Example 1: Calculate the circumference and the area and of a circle by using geometry formulas if the radius of the circle is 21 units.

To find the area and the circumference of the circle.

Given: Radius of a circle = 21 units

Using geometry formulas for circle,

Area of circle = π × r 2

= 3.142857 × 21 2

Now for the circumference of the circle,

= 2(3.142857)(21)

Answer: The area of a circle is 1385.44 square units and the circumference of a circle is 131.95 units.

Example 2: What is the area of a rectangular park whose length and breadth are 90 m and 60 m respectively?

Solution: To find the area of a rectangular park:

Given: Length of the park = 90 m

The breadth of the park = 60 m Using the geometry formulas for a rectangle,

Area of Rectangle = (Length × Breadth)

= (90 × 60) m 2

Answer: The area of the rectangular park is 5400 m 2 .

Example 3: Using geometry formulas of the cube, calculate the surface area and volume of a cube whose edge is 6 units.

Solution: To Find: The surface area and volume of a cube whose edge is 6 units

Using geometry formulas of cube, Surface area of cube is = A = 6a 2 A = 6 (6) 2 A = 6 × 36 = 216 units 2 Volume of a cube, V = a 3 V = (6) 3 V = 216 units 3

Answer: The surface area of the cube is 216 units 2 . The volume of the cube is 216 units 3

FAQs on Geometry Formulas

What are the geometry formulas of a cuboid.

The geometry formulas of a cuboid are listed below:

  • Surface Area of cuboid , A = 2(lb + bh + hl)
  • Volume of cuboid , V = lbh
  • Space diagonal of cuboid, d = √(l 2 + b 2 +h 2 )

What are the Geometry Formulas of a Rectangle?

The geometry formulas of a rectangle are listed below:

  • Perimeter of a rectangle = 2(l + w)
  • Area of rectangle = lw
  • Diagonal of a rectangle, d = √(l 2 + w 2 )
  • l = length of a rectangle
  • w = width of a rectangle

What are the Geometry Formulas of a Cone?

The geometry formulas of a cone are listed below:

  • Total surface area of cone, A = πr(r+l) = πr[r+√(h 2 +r 2 )]
  • Curved surface area of cone, A c = πrl
  • Volume of cone, V = ⅓πr 2 h
  • Slant Height of cone, l = √(h 2 +r 2 )
  • Base Area, A b = πr 2
  • r= radius of a cone
  • h= height of a cone
  • l = slant height

What are the Geometry Formulas of a Circle?

The geometry formulas of a circle are listed below:

  • Circumference = 2πr
  • Area = πr 2
  • Diameter = 2r

Where, r = radius of a circle

What are the Geometric Formulas of a Sphere?

The two important geometry formulas of a sphere are the area and volume of a sphere. The formula for the surface area of a sphere is A = 4πr 2 and the formula for the volume of the sphere is V = ⁴⁄₃πr 3 .

What are the Applications of Geometry Formulas?

Geometry formulas are useful to find the perimeter, area, volume, and surface areas of two-dimensional and 3D Geometry figures. In our day-to-day life, there are numerous objects which resemble geometric figures and the areas and volumes of these geometric figures can be calculated using these geometric formulas.

How to Learn All Geometry Formulas?

All geometry formulas are given in detail above on this page for reference. These formulas can be learnt with practice when the students use them repeatedly. Another way to memorize the geometry formulas is that the students should make a chart of all these formulas and paste it on a place or wall where they usually study. This will help them glance through the formulas more often and this will passively be absorbed by them.

Algebra: Geometry Word Problems

In these lessons, we look at geometry word problems, which involves geometric figures and angles described in words. You would need to be familiar with the formulas in geometry .

Related Pages Perimeter and Area of Polygons Nets Of 3D Shapes Surface Area Formulas Volume Formulas More Geometry Lessons

Making a sketch of the geometric figure is often helpful.

You can see how to solve geometry word problems in the following examples: Problems involving Perimeter Problems involving Area Problems involving Angles

There is also an example of a geometry word problem that uses similar triangles.

Geometry Word Problems Involving Perimeter

Example 1: A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side?

Solution: Step 1: Assign variables: Let x = length of the equal side. Sketch the figure.

Step 2: Write out the formula for perimeter of triangle . P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch. 50 = x + x + x + 5

Combine like terms 50 = 3x + 5

Isolate variable x 3x = 50 – 5 3x = 45 x =15

Be careful! The question requires the length of the third side. The length of third side = 15 + 5 =20

Answer: The length of third side is 20

Example 2: Writing an equation and finding the dimensions of a rectangle knowing the perimeter and some information about the about the length and width. The width of a rectangle is 3 feet less than its length. The perimeter of the rectangle is 110 feet. Find its dimensions.

Geometry Word Problems Involving Area

Example 1: A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Step 1: Assign variables: Let x = original width of rectangle

Step 2: Write out the formula for area of rectangle. A = lw

Step 3: Plug in the values from the question and from the sketch. 60 = (4x + 4)(x –1)

Use distributive property to remove brackets 60 = 4x 2 – 4x + 4x – 4

Put in Quadratic Form 4x 2 – 4 – 60 = 0 4x 2 – 64 = 0

This quadratic can be rewritten as a difference of two squares (2x) 2 – (8) 2 = 0

Factorize difference of two squares "> (2x) 2 – (8) 2 = 0 (2x – 8)(2x + 8) = 0

Since x is a dimension, it would be positive. So, we take x = 4

The question requires the dimensions of the original rectangle. The width of the original rectangle is 4. The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the original rectangle are 4 and 16.

Example 2: This is a geometry word problem that we can solve by writing an equation and factoring. The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. Find the height of the triangle.

Geometry Word Problems involving Angles

Example 1: In a quadrilateral two angles are equal. The third angle is equal to the sum of the two equal angles. The fourth angle is 60° less than twice the sum of the other three angles. Find the measures of the angles in the quadrilateral.

Step 1: Assign variables: Let x = size of one of the two equal angles Sketch the figure

Step 2: Write down the sum of angles in quadrilateral . The sum of angles in a quadrilateral is 360°

Step 3: Plug in the values from the question and from the sketch. 360 = x + x + (x + x) + 2(x + x + x + x) – 60

Combine like terms 360 = 4x + 2(4x) – 60 360 = 4x + 8x – 60 360 = 12x – 60

Isolate variable x 12x = 420 x = 35

The question requires the values of all the angles. Substituting x for 35, you will get: 35, 35, 70, 220

Answer: The values of the angles are 35°, 35°, 70° and 220°.

Example 2: The sum of the supplement and the complement of an angle is 130 degrees. Find the measure of the angle.

Game Central

how to solve geometry related problems

Geometry Questions

Geometry questions, with answers, are provided for students to help them understand the topic more easily. Geometry is a chapter that has been included in almost all classes. The questions will be provided in accordance with NCERT guidelines. The use of geometry can be seen in both mathematics and everyday life. Thus, the fundamentals of this topic must be understood. The questions here will cover both the fundamentals and more difficult problems for students of all levels. As a result, students will be skilled in using it to solve geometry problems. Click here to learn more about Geometry.

Here, we are going to discuss different geometry questions, based on different concepts with solutions.

Geometry Questions with Solutions

1. The lines that are equidistant from each other and never meet are called ____.

Parallel lines are the lines that are equidistant from each other and never meet. The parallel lines are represented with a pair of vertical lines and its symbol is “||”. If AB and CD are the two parallel lines, it is denoted as AB || CD.

2. If two or more points lie on the same line, they are called _____.

If two or more points lie on the same line, they are called collinear points. If points A, B and C lie on the same line “l”, then we can say that the points are collinear.

3. Find the number of angles in the following figure.

Geometry Questions - 3

In the given figure, there are three individual angles, (i.e.) 30°, 20° and 40°.

Two angles in a pair of 2. (i.e.) 20° + 30° = 50° and 20 + 40 = 60°

One angle in a pair of 3 (i.e) 20° + 30° + 40° = 90°

Hence, the total number of possible angles in the given figure is 6 .

4. In the given figure, ∠BAC = 90°, and AD is perpendicular to BC. Find the number of right triangles in the given figure.

Geometry Questions - 4

Given: ∠BAC = 90° and AD⊥BC.

Since AD⊥BC, the two possible right triangles obtained are ∠ADB and ∠ADC.

Hence, the number of right triangles in the given figure is 3.

I.e., ∠BAC = ∠ADB = ∠ADC = 90°.

5. The length of a rectangle is 3 more inches than its breadth. The area of the rectangle is 40 in 2 . What is the perimeter of the rectangle?

Given: Area = 40 in 2 .

Let “l” be the length and “b” be the breadth of the rectangle.

According to the given question,

b = b and l = 3+b

We know that the area of a rectangle is lb units.

So, 40 = (3+b)b

40 = 3b +b 2

This can be written as b 2 +3b-40 = 0

On factoring the above equation, we get b= 5 and b= -8.

Since the value of length cannot be negative, we have b = 5 inches.

Substitute b = 5 in l = 3 + b, we get

l = 3 + 5 = 8 inches.

As we know, the perimeter of a rectangle is 2(l+b) units

P = 2 ( 8 + 5)

P = 2 (13) = 26

Hence, the perimeter of a rectangle is 26 inches.

6. What is the area of a circle in terms of π, whose diameter is 16 cm?

Given: Diameter = 16 cm.

Hence, Radius, r = 8 cm

We know that the area of a circle = πr 2 square units.

Now, substitute r = 8 cm in the formula, we get

A = π(8) 2 cm 2

A = 64π cm 2

Hence, the area of a circle whose diameter is 16 cm = 64π cm 2 .

7. Find the missing angle in the given figure.

Geometry Questions - 7

Given two angles are 35° and 95°.

Let the unknown angle be “x”.

We know that sum of angles of a triangle is 180°

Therefore, 35°+95°+x = 180°

130°+ x = 180°

x = 180° – 130°

Hence, the missing angle is 50°.

8. Find the curved surface area of a hemisphere whose radius is 14 cm.

Given: Radius = 14 cm.

As we know, the curved surface area of a hemisphere is 2πr 2 square units.

CSA of hemisphere = 2×(22/7)×14×14

CSA = 2×22×2×14

Hence, the curved surface area of a hemisphere is 1232 cm 2 .

9. Find the volume of a cone in terms π, whose radius is 3 cm and height is 4 cm.

Given: Radius = 3 cm

Height = 4 cm

We know that the formula to find the volume of a cone is V = (⅓)πr 2 h cubic units.

Now, substitute the values in the formula, we get

V = (⅓)π(3) 2 (4)

V = π(3)(4)

V = 12π cm 3

Hence, the volume of a cone in terms of π is 12π cm 3 .

10. The base area of a cylinder is 154 cm 2 and height is 5 cm. Find the volume of a cylinder.

Given: Base area of a cylinder = 154 cm 2 .

As the base area of a cylinder is a circle, we can write πr 2 = 154cm 2 .

We know that the volume of a cylinder is πr 2 h cubic units.

V = 154(5) cm 3

V = 770 cm 3

Hence, the volume of a cylinder is 770 cm 2 .

Practice Questions

  • Find the area of a square whose side length is 6 cm.
  • Find the number of obtuse angles in the given figure.

Geometry Questions - Practice 2

            3. Find the number of line segments in the given figure and name them.

Geometry Questions - Practice 3

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What are 'multiplication facts'? Why are they essential to your child's success in math?

by Bronwyn Reid O'Connor and Benjamin Zunica, The Conversation

multiplication

One of the essential skills students need to master in primary school mathematics are "multiplication facts."

What are they? What are they so important? And how can you help your child master them?

What are multiplication facts?

Multiplication facts typically describe the answers to multiplication sums up to 10x10. Sums up to 10x10 are called "facts" as it is expected they can be easily and quickly recalled. You may recall learning multiplication facts in school from a list of times tables.

The shift from "times tables" to "multiplication facts" is not just about language. It stems from teachers wanting children to see how multiplication facts can be used to solve a variety of problems beyond the finite times table format.

For example, if you learned your times tables in school (which typically went up to 12x12 and no further), you might be stumped by being asked to solve 15x8 off the top of your head. In contrast, we hope today's students can use their multiplication facts knowledge to quickly see how 15x8 is equivalent to 10x8 plus 5x8.

The shift in terminology also means we are encouraging students to think about the connections between facts. For example, when presented only in separate tables, it is tricky to see how 4x3 and 3x4 are directly connected.

Math education has changed

In a previous piece, we talked about how mathematics education has changed over the past 30 years.

In today's mathematics classrooms, teachers still focus on developing students' mathematical accuracy and fast recall of essential facts, including multiplication facts.

But we also focus on developing essential problem-solving skills. This helps students form connections between concepts, and learn how to reason through a variety of real-world mathematical tasks.

Why are multiplication facts so important?

By the end of primary school, it is expected students will know multiplication facts up to 10x10 and can recall the related division fact (for example, 10x9=90, therefore 90÷10=9).

Learning multiplication facts is also essential for developing "multiplicative thinking." This is an understanding of the relationships between quantities, and is something we need to know how to do on a daily basis.

When we are deciding whether it is better to purchase a 100g product for $3 or a 200g product for $4.50, we use multiplicative thinking to consider that 100g for $3 is equivalent to 200g for $6—not the best deal!

Multiplicative thinking is needed in nearly all math topics in high school and beyond. It is used in many topics across algebra, geometry, statistics and probability.

This kind of thinking is profoundly important. Research shows students who are more proficient in multiplicative thinking perform significantly better in mathematics overall.

In 2001, an extensive RMIT study found there can be as much as a seven-year difference in student ability within one mathematics class due to differences in students' ability to access multiplicative thinking.

These findings have been confirmed in more recent studies, including a 2021 paper .

So, supporting your child to develop their confidence and proficiency with multiplication is key to their success in high school mathematics. How can you help?

Below are three research-based tips to help support children from Year 2 and beyond to learn their multiplication facts.

1. Discuss strategies

One way to help your child's confidence is to discuss strategies for when they encounter new multiplication facts.

Prompt them to think of facts they already and how they can be used for the new fact.

For example, once your child has mastered the x2 multiplication facts, you can discuss how 3x6 (3 sixes) can be calculated by doubling 6 (2x6) and adding one more 6. We've now realized that x3 facts are just x2 facts "and one more"!

What are 'multiplication facts'? Why are they essential to your child's success in maths?

Strategies can be individual: students should be using the strategy that makes the most sense to them. So you could ask a questions such as "if you've forgotten 6x7, how could you work it out?" (we might personally think of 6x6=36 and add one more 6, but your child might do something different and equally valid).

This is a great activity for any quiet car trip. It can also be a great drawing activity where you both have a go at drawing your strategy and then compare. Identifying multiple strategies develops flexible thinking.

2. Help them practice

Practicing recalling facts under a friendly time crunch can be helpful in achieving what teachers call "fluency" (that is, answering quickly and easily).

A great game you could play with your children is " multiplication heads up " . Using a deck of cards, your child places a card to their forehead where you can see but they cannot. You then flip over the top card on the deck and reveal it to your child. Using the revealed card and the card on your child's head you tell them the result of the multiplication (for example, if you flip a 2 and they have a 3 card, then you tell them "6!").

Based on knowing the result, your child then guesses what their card was.

If it is challenging to organize time to pull out cards, you can make an easier game by simply quizzing your child. Try to mix it up and ask questions that include a range of things they know well with and ones they are learning.

Repetition and rehearsal will mean things become stored in long-term memory.

3. Find patterns

Another great activity to do at home is print some multiplication grids and explore patterns with your child.

What are 'multiplication facts'? Why are they essential to your child's success in maths?

A first start might be to give your child a blank or partially blank multiplication grid which they can practice completing.

Then, using colored pencils, they can color in patterns they notice. For example, the x6 column is always double the answer in the x3 column. Another pattern they might see is all the even answers are products of 2, 4, 6, 8, 10. They can also notice half of the grid is repeated along the diagonal.

This also helps your child become a mathematical thinker, not just a calculator.

The importance of multiplication for developing your child's success and confidence in mathematics cannot be understated. We believe these ideas will give you the tools you need to help your child develop these essential skills.

Provided by The Conversation

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Praxis Core Math

Course: praxis core math   >   unit 1.

  • Algebraic properties | Lesson
  • Algebraic properties | Worked example
  • Solution procedures | Lesson
  • Solution procedures | Worked example
  • Equivalent expressions | Lesson
  • Equivalent expressions | Worked example
  • Creating expressions and equations | Lesson
  • Creating expressions and equations | Worked example

Algebraic word problems | Lesson

  • Algebraic word problems | Worked example
  • Linear equations | Lesson
  • Linear equations | Worked example
  • Quadratic equations | Lesson
  • Quadratic equations | Worked example

What are algebraic word problems?

What skills are needed.

  • Translating sentences to equations
  • Solving linear equations with one variable
  • Evaluating algebraic expressions
  • Solving problems using Venn diagrams

How do we solve algebraic word problems?

  • Define a variable.
  • Write an equation using the variable.
  • Solve the equation.
  • If the variable is not the answer to the word problem, use the variable to calculate the answer.

What's a Venn diagram?

  • Your answer should be
  • an integer, like 6 ‍  
  • a simplified proper fraction, like 3 / 5 ‍  
  • a simplified improper fraction, like 7 / 4 ‍  
  • a mixed number, like 1   3 / 4 ‍  
  • an exact decimal, like 0.75 ‍  
  • a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
  • (Choice A)   $ 4 ‍   A $ 4 ‍  
  • (Choice B)   $ 5 ‍   B $ 5 ‍  
  • (Choice C)   $ 9 ‍   C $ 9 ‍  
  • (Choice D)   $ 14 ‍   D $ 14 ‍  
  • (Choice E)   $ 20 ‍   E $ 20 ‍  
  • (Choice A)   10 ‍   A 10 ‍  
  • (Choice B)   12 ‍   B 12 ‍  
  • (Choice C)   24 ‍   C 24 ‍  
  • (Choice D)   30 ‍   D 30 ‍  
  • (Choice E)   32 ‍   E 32 ‍  
  • (Choice A)   4 ‍   A 4 ‍  
  • (Choice B)   10 ‍   B 10 ‍  
  • (Choice C)   14 ‍   C 14 ‍  
  • (Choice D)   18 ‍   D 18 ‍  
  • (Choice E)   22 ‍   E 22 ‍  

Things to remember

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So Apparently We Just Impeach Cabinet Members for Fun Now?

This is Totally Normal Quote of the Day , a feature highlighting a statement from the news that exemplifies just how extremely normal everything has become.

“Who said it was gonna fix the problem?” —Republican Rep. Ralph Norman, when an MSNBC reporter asked him how impeaching DHS Secretary Alejandro Mayorkas would solve the problems at the U.S. southern border

If at first you don’t succeed, try, try again. That’s how House Republicans managed—barely—to impeach Homeland Security Secretary Alejandro Mayorkas on Tuesday night, in their second attempt this month.

In a 214–213 vote, House Republicans made history by impeaching the first sitting Cabinet official in 148 years. (Secretary of War William Belknap was impeached most recently , in 1876.) House Majority Leader Steve Scalise came back to Capitol Hill to vote for impeachment after receiving a round of treatment for blood cancer. Meanwhile, two Democrats—Reps. Lois Frankel and Judy Chu—were absent, and presumably if either one had been present to vote, Mayorkas would not have been impeached.

“Desperate times call for desperate measures,” Speaker Mike Johnson said in a press conference Wednesday morning. “We had to do that.”

Did they really have to, though? As recently as one week ago, House Republicans were not so sure impeachment was necessary, and fell one vote short in impeaching Mayorkas. Shortly before the first vote, Rep. Mike Gallagher, who crossed party lines and voted no, criticized the party’s motivations for pursuing impeachment in a Wall Street Journal op-ed, noting that there were no actual criminal offenses cited, only underenforcement of current immigration policies. “If we are to make underenforcement of the law, even egregious underenforcement, impeachable, almost every cabinet secretary would be subject to impeachment,” wrote Gallagher, who also just announced he’s not seeking reelection .

“Creating a new, lower standard for impeachment, one without any clear limiting principle, wouldn’t secure the border or hold Mr. Biden accountable,” Gallagher added. “It would only pry open the Pandora’s box of perpetual impeachment.”

And in the same week of the first Mayorkas impeachment vote, Republicans blew up their very own immigration bill , legislation that would have beefed up border security and sent more aid to Ukraine. Why? Largely because presumptive Republican presidential nominee Donald Trump declared he needed chaos at the southern border to continue for the sake of his campaign .

Mayorkas’ impeachment is now in the hands of the Senate, which will most likely dismiss the charges against him, allowing the homeland security secretary to resume his regular duties. Then what was all this for? Well, Norman might have just admitted the quiet part out loud.

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IMAGES

  1. How to solve geometry problems

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  2. Using Algebra to Solve Geometry Problems

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  3. Using Algebra to solve Geometry Problems involving angles.

    how to solve geometry related problems

  4. How to solve geometry problems? part 2

    how to solve geometry related problems

  5. help solving geometry problems

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  6. Circle Theorems

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VIDEO

  1. 10 Challenging Geometry Problems

  2. Can you solve?

  3. How to Solve Geometry Question with Shortcuts by Karunagaran

  4. How to solve this equation in geometric ways?

  5. Solve Geometry Problems by using 2 Variables

  6. how to solve Geometry question quickly ? #maths #mathematics #mathshorts #mathstricks

COMMENTS

  1. Geometry Math Problems (solutions, examples, videos, examples)

    Solution: Step 1: Assign variables: Let x = length of the equal sides Sketch the figure Step 2: Write out the formula for perimeter of triangle. P = sum of the three sides Step 3: Plug in the values from the question and from the sketch. 50 = x + x + x+ 5 Combine like terms 50 = 3x + 5 Isolate variable x 3x = 50 - 5 3x = 45 x = 15 Be careful!

  2. Free Geometry Problems and Questions writh Solutions

    Find the overlapping area of two circles: problem with detailed solutions. Sectors and Circles Problems. Problems, with detailed solutions, related to sectors and circles. Two Squares and a Circle - Problem With Solution. A problem, with a detailed solution, on a circle inscribed in one square and circumscribed to another, is presented.

  3. Equations and geometry

    Algebra basics 8 units · 112 skills. Unit 1 Foundations. Unit 2 Algebraic expressions. Unit 3 Linear equations and inequalities. Unit 4 Graphing lines and slope. Unit 5 Systems of equations. Unit 6 Expressions with exponents. Unit 7 Quadratics and polynomials. Unit 8 Equations and geometry.

  4. Using Geometric Concepts and Properties to Solve Problems

    Step 2: Make a plan. draw a picture. look for a pattern. systematic guessing and checking. acting it out. making a table. working a simpler problem. working backwards. Step 3: Implement your plan.

  5. Solve

    Graph GCF LCM New Example Help Tutorial Solve an equation, inequality or a system. Example: 2x-1=y,2y+3=x What can QuickMath do? QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

  6. How to Solve Practical Geometry Problems

    1. Determine what you need to calculate to solve the problem. In some cases, you may need a length; in others, an area or angle measure. If you are conscious throughout the process of what you need to determine, you can save yourself a significant amount of time. 2. Draw a diagram.

  7. All Geometry Formulas

    Let us have a look at solved examples to understand the basic geometry formulas. Solved Examples Using Geometry Formulas. Example 1: Calculate the circumference and the area and of a circle by using geometry formulas if the radius of the circle is 21 units. Solution: To find the area and the circumference of the circle.

  8. Similarity

    Test your understanding of Similarity with these % (num)s questions. Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems and solve some problems with polygons.

  9. Circles

    Unit 1 Lines Unit 2 Angles Unit 3 Shapes Unit 4 Triangles Unit 5 Quadrilaterals Unit 6 Coordinate plane Unit 7 Area and perimeter Unit 8 Volume and surface area Unit 9 Pythagorean theorem Unit 10 Transformations Unit 11 Congruence Unit 12 Similarity Unit 13 Trigonometry Unit 14 Circles Unit 15 Analytic geometry Unit 16 Geometric constructions

  10. Algebra: Geometry Word Problems

    Step 1: Assign variables: Let x = length of the equal side. Sketch the figure. Step 2: Write out the formula for perimeter of triangle. P = sum of the three sides Step 3: Plug in the values from the question and from the sketch. 50 = x + x + x + 5 Combine like terms 50 = 3x + 5 Isolate variable x 3x = 50 - 5 3x = 45 x =15 Be careful!

  11. Similarity example problems

    Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-...

  12. Step-by-Step Calculator

    Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem. Is there a step by step calculator for physics?

  13. Microsoft Math Solver

    Microsoft Math Solver - Math Problem Solver & Calculator Type a math problem Solve trigonometry Get step-by-step explanations See how to solve problems and show your work—plus get definitions for mathematical concepts Graph your math problems Instantly graph any equation to visualize your function and understand the relationship between variables

  14. Geometry Calculator

    Circle Parallels Interactive geometry calculator. Create diagrams, solve triangles, rectangles, parallelograms, rhombus, trapezoid and kite problems.

  15. Solve

    Integration. ∫ 01 xe−x2dx. Limits. x→−3lim x2 + 2x − 3x2 − 9. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

  16. Using Geometry to Analyze Math Problems

    A Math Problem. Geometry is the field of mathematics that deals with points, lines, and shapes. Once you've learned some geometry, you'll be able to solve many real world problems. Geometry skills ...

  17. Isosceles & equilateral triangles problems (video)

    AboutTranscript. Isosceles triangles have two congruent sides and two congruent base angles. Equilateral triangles have all side lengths equal and all angle measures equal. We use these properties to find missing angles in composite figures. The problems are partly from Art of Problem Solving, by Richard Rusczyk.

  18. Geometry Questions

    Solution: Parallel lines are the lines that are equidistant from each other and never meet. The parallel lines are represented with a pair of vertical lines and its symbol is "||". If AB and CD are the two parallel lines, it is denoted as AB || CD. 2. If two or more points lie on the same line, they are called _____. Solution:

  19. Solving geometry Problems

    1 Answer. Sorted by: 4. If you want to solve an Olympiad problem, especially geometric one, you'll have to pracitce a lot. Practicing and solving a lot of geometric problems will make you familiar with all these theoremas and their application. Also that can better your creativity, because when you check one problem's solution (even if you don ...

  20. Mathway

    Calculus Free math problem solver answers your calculus homework questions with step-by-step explanations.

  21. What are 'multiplication facts'? Why are they essential to your child's

    Related Stories. Mathematical bedtime stories may build better mathematical memory. Oct 3, 2023. Why teachers are letting students solve math problems in lots of different ways. Feb 1, 2023.

  22. Unknown angle problems (with algebra) (practice)

    Finding missing angles Find measure of angles word problem Equation practice with complementary angles Math > 7th grade > Geometry > Missing angle problems Unknown angle problems (with algebra) Google Classroom Solve for x in the diagram below. x ∘ ( 3 x + 10) ∘ x = ∘ Stuck? Review related articles/videos or use a hint. Report a problem

  23. Math Message Boards FAQ & Community Help

    contests on aops Practice Math Contests USABO news and information AoPS Blog Emergency Homeschool Resources Podcast: Raising Problem Solvers just for fun Reaper Greed Control All Ten

  24. Some Texas schools try new way to teach math to students

    DALLAS — In Eran McGowan's math class, students try to teach each other. If a student is brave enough to share how they solved a math problem, they stand up in front of the other third graders ...

  25. Algebraic word problems

    Solving algebraic word problems requires us to combine our ability to create equations and solve them. To solve an algebraic word problem: Define a variable. Write an equation using the variable. Solve the equation. If the variable is not the answer to the word problem, use the variable to calculate the answer.

  26. Why was Alejandro Mayorkas impeached? Not to solve any problems

    "Who said it was gonna fix the problem?" —Republican Rep. Ralph Norman, when an MSNBC reporter asked him how impeaching DHS Secretary Alejandro Mayorkas would solve the problems at the U.S ...

  27. Mathway

    Free math problem solver answers your algebra homework questions with step-by-step explanations.