OnTrack logo

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.

solving mathematical problems involving two dimensional figures

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?

Copy and paste the link code above.

Related Items

2 dimensional shapes

2-Dimensional Shapes and Their Properties

In grade 5, students learn about the different 2-dimensional shapes, such as circles, squares, triangles, etc. They gain an in-depth understanding of what these shapes represent, and learn how to identify their properties.

Lessons on 2-dimensional shapes can be lots of fun if math teachers are equipped with the right resources! To this end, we bring you a few teaching ideas and awesome activities on this topic. Use them in your class and see students’ math knowledge soar in no time!

solving mathematical problems involving two dimensional figures

Ideas for Teaching 2-Dimensional Shapes and Their Properties

What are 2-dimensional shapes.

For starters, define what 2-dimensional shapes are. Explain that a 2-dimensional shape (or a 2-D shape) is a shape that only has two measurable dimensions – length and width. It is a flat plane figure that has no depth or thickness.

By now, students have come across many 2-dimensional shapes. Add that there are many different 2-dimensional shapes and draw examples of such shapes on the whiteboard. If you have manipulatives or images, even better. You can include shapes such as the following:

2 dimensional shapes

Students have also encountered 3-dimensional shapes. You can draw a distinction between the two to make things clearer. While 2-dimensional shapes are flat figures with length and width, 3-dimensional shapes are solid figures with length, width, and height.

Draw an example on the whiteboard where there are both 2-dimensional figures and 3-dimensional figures and ask students to circle the 2-dimensional ones. This could look something like this:

2 dimensional shapes

If students understand the difference between 2-dimensional and 3-dimensional shapes, they should be able to recognize that the cylinder and rectangular prism have height in addition to length and width, and are not flat, hence they’re not 2-dimensional objects.

Properties of 2-Dimensional Figures

Point out that there are many 2-dimensional figures around us and that each of them has its own unique properties. These properties make each 2-dimensional figure special, while there are also some that share common characteristics.

Add that one example of 2-D figures are polygons. Explain that a polygon is a closed plane figure, formed by connecting segments that are called ‘sides’. It’s a 2-dimensional figure whose name is based on the number of its sides. Thus, we have triangles, squares, rectangles, etc.

Ask students to reflect on the word polygon. Point out that in Greek, ‘poly’ means ‘many’, whereas ‘gon’ means ‘angle’. So polygon means ‘many angles’, and this refers to the angles formed by the straight lines (or sides) of the polygon.

Add that a polygon always has straight sides – if the figure has one side that has a curve, it does not represent a polygon. Also, polygons are always closed shapes – if the figure is open, it does not represent a polygon.

Draw a few figures on the whiteboard, and make sure to include shapes that are polygons and shapes that aren’t. Ask students to circle the ones that are polygons and explain why these qualify as polygons. Then, provide examples of polygons and their properties.

Example 1 – Triangle:

Draw a triangle on the whiteboard and highlight that a triangle is a 2-dimensional figure that has 3 sides that may or may not be equal. Every triangle has 3 vertices, 3 angles that may or may not be equal:

Triangle

Example 2 – Quadrilateral:

Draw a quadrilateral on the whiteboard and explain that a quadrilateral is a 2-dimensional figure that has 4 sides and 4 vertices. Add that a quadrilateral is just a flat figure with four sides, all of which connect up and are straight.

Quadrilateral

Example 3 – Parallelogram:

Draw a parallelogram and point out that this is a quadrilateral and thus a 2-D figure. It has 4 sides, 4 vertices, 2 pairs of opposite sides that are parallel. Every parallelogram also has 2 pairs of opposite sides that are equal, as well as 2 pairs of opposite angles that are equal.

Parallelogram

Example 4 – Square:

Draw a square and point out that it falls into the category of quadrilaterals. Add that a square is a 2-dimensional figure that has 4 equal sides, 4 vertices, 2 pairs of parallel sides, and 4 right angles.

Square

Example 5 – Rectangle:

Draw a rectangle and add that this is a type of quadrilateral. It’s a 2-D figure that has 2 pairs of opposite sides that are congruent, as well as 4 vertices and 2 pairs of parallel sides. Every rectangle has 4 right angles.

Rectangle

Example 5 – Rhombus:

Draw a rhombus on the whiteboard. Add that this is a 2-dimensional figure that has 4 equal sides (a quadrilateral), 4 vertices, 2 pairs of parallel sides, and 2 pairs of opposite angles that are equal.

Rhombus

Example 6 – Trapezoid:

Draw a trapezoid and add that this is a quadrilateral. It’s a 2-dimensional figure that has 4 sides and 4 vertices. It also has one pair of parallel sides. It can have a pair of equal sides, pairs of equal angles, as well as right angles.

Trapezoid

Example 7 – Kite:

Draw a kite on the whiteboard, which also falls into the category of quadrilateral figures. Explain that a kite is a 2-dimensional figure that has 4 sides and 4 vertices. A kite has no pair of parallel sides, and it has 2 pairs of consecutive sides that are equal.

Kite

Example 8 – Pentagon

Draw a pentagon on the whiteboard. Point out that a pentagon is a 2-dimensional figure that has 5 sides that can be equal, and also has 5 angles that can be equal.

Pentagon

Example 9 – Hexagon

Draw a hexagon on the whiteboard and explain that this is a 2-dimensional figure that has 6 sides that can be equal, as well as 6 angles that can be equal.

Hexagon

Example 10 – Heptagon

Draw an example of a heptagon on the whiteboard. Point out that a heptagon is a 2-dimensional figure that has 7 sides that can be equal, in addition to 7 angles that can be equal.

Heptagon

Example 11 – Octagon

Draw an example of an octagon on the whiteboard. Explain that this is a 2-dimensional figure that has 8 sides that can be equal. It also has 8 angles that can be equal.

Octagon

Types of Polygons

Explain that 2-dimensional figures such as polygons are classified into two categories: regular and irregular polygons. All the sides and angles or regular polygons are equal, while irregular polygons have unequal sides and angles. Draw examples of the two:

solving mathematical problems involving two dimensional figures

Point out that in the first figure, all sides and angles are equal, so this is a regular polygon. In the second figure, on the other hand, the sides and angles are unequal, which makes the second figure an irregular polygon.

Add that there are a lot more polygons on the list, and as mentioned, each polygon is named according to the number of its sides.

Is a Circle a Polygon?

Draw a circle on the whiteboard and ask students to observe it and to reflect whether this figure represents a polygon.

Circle

Explain that a circle is a closed plane figure. It is not a polygon because it doesn’t have corners or edges. It has a center whose distance to the edge of the circle is always the same. Add that although a circle is not a polygon, it’s still a 2-dimensional shape.

Additional Resources:

If you have the technical means in your classroom, you could also enrich your lesson on 2-dimensional shapes and their properties with multimedia material, such as videos. This is especially useful for illustrating the variety of figures.

For instance, use this video as an introduction to 2-dimensional shapes, and to the topic of what having 2 dimensions actually means. Then, play this video on recognizing the different shapes, such as circle, triangle, square, parallelogram, etc.

This video is a great resource for illustrating where the shapes exist in everyday life, which is achieved through a fun song on shapes and their properties. In addition, this video that also consists of a song called ‘Come and Meet the 2-D Shapes’ is guaranteed to make your lesson more amusing.

Activities to Practice 2-Dimensional Shapes

Compare shapes game.

This is a simple online game developed by Khan Academy where students will practice their knowledge of the different 2-D shapes and their properties. To use the game as an activity in your class, make sure there are enough devices for the whole class.

Divide students into pairs and provide instructions for the game. Point out that in each exercise in the game, students have to compare two given shapes and determine which one fulfills the properties in the specific question.

In the end, students in each pair compare their results and discuss their answers. The person with the highest score in each pair wins the game. Homeschooling parents can adjust the game to an individual one.

Properties of Shapes Game

In this brief online game by Khan Academy, students get to apply their knowledge of properties of 2-dimensional shapes. To implement the game in your class, provide suitable devices for students.

Arrange students in pairs. The two students in each pair play together and try to answer the questions correctly. In each exercise, they have to identify all answers that are true for a given statement on a 2-dimensional figure.

By doing so, the game provides a peer-tutoring approach as well. If students get stuck, they can also use a hint in the game or watch a related video for help. Parents who are homeschooling can also use the game for individual practice.

2-D and 3-D Shape Sort: Factory

This is a fun online game where students practice differentiating between 2-dimensional figures as flat objects with 2 dimensions and 3-dimensional figures as solid objects with 3 dimensions. The only materials needed for the game are suitable devices.

Students play the game individually. Provide instructions. Point out that in the game, students are asked to help Muggo clean up and sort his shapes into two boxes: one of which is a 2-D box and the other one a 3-D box.

Once students are done sorting the shapes into the right box, ask them to share their thoughts on the exercise. How did they know which shape goes into which box? Which dimensions do 2-D shapes have and which ones do 3-D shapes have?

Quadrilaterals Interactive Game

In this online game on 2-D shapes, students get to use their skills at recognizing the types of quadrilaterals, as well as their properties. Provide a suitable device for each student and play the game!

Arrange students in pairs. Provide instructions for the game. Explain that in each exercise, students are given an image of a 2-dimensional figure and a few descriptions. Players have to select all descriptive words that apply to the given image.

This is a fast-paced game, so students should aim to answer as many questions as possible and as quickly as possible. In the end, the two players in each pair compare their final scores. The person with the highest score wins the game.

‘What Am I?’ Game

This is a fun guessing game that is bound to get students excited about 2-d shapes. To implement this game in your class, you’ll need construction paper, some scissors, and some markers.

Draw task cards on the construction paper and write a question on guessing a 2-d shape on each task card. For instance, you can create a question such as: ‘I have four sides and four angles, but my sides are never parallel. What am I?’

Include the answer under the question on each task card as well. For example, in the above case, you would include the word ‘kite’ under the question. Create as many task cards as necessary, depending on the size of your class.

Divide students into pairs and place the task cards in the middle, face down. Use at least 20 cards per pair. Provide instructions for the game. Payer 1 draws a card and reads the question to player 2. Player 2 answers the question.

If the answer is correct, they score 1 point. If the answer is incorrect, they lose 2 points. Player 2 then repeats the procedure by drawing a new card with a question for player 1. Keep playing until the pair runs out of cards.

At this moment, students calculate their final scores and decide on the winner. You can also include a symbolic prize for the person that wins the game in each pair, such as not having to do homework all week long.

Before You Leave…

If you enjoyed these ideas and activities for teaching 2-dimensional shapes and their properties, you’ll want to check out our lesson that’s dedicated to this topic! So if you need guidance to structure your class and teach it, sign up for our emails to receive loads of free content!

Feel free to also check out our blog – you’ll find plenty of awesome resources that you can use in your class, such as this awesome article with free worksheets and activities on classifying 2-dimensional shapes!

And if you’re ready to become a member, simply join our Math Teacher Coach community!

This article is based on:

Unit 7 – Geometry

  • 7-1 Exploring the Coordinate System
  • 7-2 Points in the Coordinate Plane
  • 7-3 Drawing Figures in the Coordinate Plane
  • 7-4 Properties of 2-Dimensional Shapes
  • 7-5 Classifying 2-Dimensional Shapes
  • 7-6 Introduction to Quadrilaterals
  • 7-7 Classifying Quadrilaterals

Share this:

  • Click to share on Twitter (Opens in new window)
  • Click to share on Facebook (Opens in new window)

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Notify me of follow-up comments by email.

Notify me of new posts by email.

Curriculum  /  Math  /  7th Grade  /  Unit 6: Geometry  /  Lesson 16

Lesson 16 of 21

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Identify and describe two-dimensional figures that result from slicing three-dimensional figures.

Common Core Standards

Core standards.

The core standards covered in this lesson

7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Foundational Standards

The foundational standards covered in this lesson

5.G.B.3 — Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Measurement and Data

5.MD.C.3 — Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

  • Understand that a three-dimensional figure can be sliced in various ways that result in a two-dimensional cross-section . 
  • Understand that a slice through a right pyramid or prism, parallel to the base, will have the same shape as the base.
  • Identify the two-dimensional figures that result from various slices through right pyramids and prisms. 

Suggestions for teachers to help them teach this lesson

Visualizing the two-dimensional figures that result from slicing a three-dimensional figure can be challenging for some students. The following are two visual supports:

  • This GeoGebra applet  Sections of Prisms and Cylinders  is a great interactive tool. The slicing plane can be moved around to show the different two-dimensional figures that are created.  
  • This Math Shorts episode  Slicing Three Dimensional Figures is a good, quick introduction as well.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Justine has a block of cheese in the shape of a square prism, as shown below.

solving mathematical problems involving two dimensional figures

She cuts the block in half. What are two different two-dimensional faces that could result from Justine slicing the cheese in half?

Guiding Questions

A right rectangular pyramid is shown below.

solving mathematical problems involving two dimensional figures

Describe how the following two-dimensional figures can be created by slicing the pyramid. 

a.   rectangle

b.   triangle

c.   trapezoid

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A cube is sliced with a single straight cut, creating a two-dimensional cross-section. Name 2 different two-dimensional shapes that could result from the slice, and explain or draw how they are created.

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • EngageNY Mathematics Grade 7 Mathematics > Module 6 > Topic C > Lesson 16 — Examples, Exercises, Problem Set
  • Illustrative Mathematics Cube Ninjas!
  • MARS Formative Assessment Lessons for High School Representing 3D Objects in 2D
  • EngageNY Mathematics Grade 7 Mathematics > Module 6 > Topic C > Lesson 17 — Examples, Exercises, Problem Set

Topic A: Angle Relationships

Identify and determine values of angles in complementary and supplementary relationships.

Use vertical, complementary, and supplementary angle relationships to find missing angles. 

Use equations to solve for unknown angles. (Part 1)

Use equations to solve for unknown angles. (Part 2)

Create a free account to access thousands of lesson plans.

Already have an account? Sign In

Topic B: Circles

Define circle and identify the measurements radius, diameter, and circumference. 

Determine the relationship between the circumference and diameter of a circle and use it to solve problems. 

Solve real-world and mathematical problems using the relationship between the circumference of a circle and its diameter. 

Determine the relationship between the area and radius of a circle and use it to solve problems.

Solve real-world and mathematical problems using the relationship between the area of a circle and its radius.

Solve problems involving area and circumference of two-dimensional figures (Part 1).

7.G.B.4 7.G.B.6

Solve problems involving area and circumference of two-dimensional figures (Part 2).

Topic C: Building Polygons and Triangles

Draw two-dimensional geometric shapes using rulers, protractors, and compasses. 

7.G.A.2 7.G.B.5

Determine if three side lengths will create a unique triangle or no triangle. 

Identify unique and identical triangles. 

Determine if conditions describe a unique triangle, no triangle, or more than one triangle.

Topic D: Solid Figures

Find the surface area of right prisms.

 Find the surface area of right pyramids.

Find the volume of right prisms and pyramids.

Solve real-world and mathematical problems involving volume.

Distinguish between and solve real-world problems involving volume and surface area.

Request a Demo

See all of the features of Fishtank in action and begin the conversation about adoption.

Learn more about Fishtank Learning School Adoption.

Contact Information

School information, what courses are you interested in, are you interested in onboarding professional learning for your teachers and instructional leaders, any other information you would like to provide about your school.

Effective Instruction Made Easy

Effective Instruction Made Easy

Access rigorous, relevant, and adaptable math lesson plans for free

solving mathematical problems involving two dimensional figures

Similar Figures

Similar figures mean when two figures are of the same shape but are of different sizes. In other words, two figures are called similar when they both have a lot of the same properties but still may not be identical. For example, the sun and moon might appear the same size but they are actually different in size. However, we are similar figures since both the figures are circular in nature. This phenomenon is considered as the property of similarity keeping the shape and distance in mind. Let us learn more about this interesting concept by defining similar figures, their role in geometry, and solve a few examples.

Similar Figures Definition

When two or more objects or figures appear the same or equal due to their shape, this property is known as a similarity or similar figures. When we magnify or demagnify these figures, they always superimpose each other. In geometry, when two shapes such as triangles, polygons, quadrilaterals, etc have the same dimension or common ratio but size or length is different, they are considered similar figures. For example, two circles (of any radii) are of the same shape but different sizes because they are similar. Look at the image below.

Similar Figures

The symbol to express similar figures is the same symbol for congruence i.e. "∼" but similar does not mean the same in size. Shapes are also considered to be similar when the ratios of the corresponding sides are equivalent i.e. while dividing each set of corresponding side lengths, the number derived is the scale factor . This number helps in increasing or decreasing the figures in size but not in shape leaving them looking like similar figures. For example, a rectangle has a length of 5 units and a width of 2 units. Now, if we increase the size of this rectangle by a scale factor of 2, the sides will become 10 units and 4 units, respectively. Hence, we can use the scale factor to get the dimensions of the changed figures.

Application of Similar Figures

Some applications of similarity or similar figures are mentioned below.

  • The similarity is widely used in Architecture.
  • Solving problems involves height and distance.
  • Solving Mathematical problems involving triangles.

Similarity of Triangles

Two triangles will be similar if the angles are equal ( corresponding angles ) and sides are in the same ratio or proportion( corresponding sides ). Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides or scale factor must be the same. If two triangles are similar that means,

  • All corresponding angle pairs of triangles are equal.
  • All corresponding sides of triangles are proportional.

Let us understand the similarity of triangles with the three theorems according to their angles and sides.

AA Similarity Criterion

The AA criterion for triangle similarity states that if the three angles of one triangle are respectively equal to the three angles of the other, then the two triangles will be similar. In short, equiangular triangles are similar. Ideally, the name of this criterion should then be the AAA(Angle-Angle-Angle) criterion, but we call it as AA criterion because we need only two pairs of angles to be equal - the third pair will then automatically be equal by the angle sum property of triangles.

Consider the following figure, in which ΔABC and ΔDEF are equi-angular,i.e.,

Using the AA criterion, we can say that these triangles are similar.

AA Similarity Criterion

SSS Similarity Criterion

The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar. This essentially means that any such pair of triangles will be equiangular(All corresponding angle pairs are equal) also. Consider the following figure, in which the sides of two triangles ΔABC and ΔDEF are respectively proportional:

SSS Similarity Criterion

That is, it is given that:

\[\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}\]

SAS Similarity Criterion

The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar. Note the emphasis on the word included. If the equal angle is a non-included angle, then the two triangles may not be similar. Consider the following figure:

SAS Similarity Criterion

It is given that

\[\begin{align}& \frac{{AB}}{{DE}} = \frac{{AC}}{{DF}} \end{align}\]

The SAS criterion tells us that ΔABC ~ ΔDEF

Similarity of Polygons

Similar polygons have the same shape, whereas their sizes are different. There would be certain uniform ratios in similar polygons. In other words, the corresponding angles are congruent, but the corresponding sides are proportional. Polygons are two-dimensional shapes composed of straight lines. They are said to have a closed shape as all the lines are connected. There are two crucial properties of similar polygons:

  • The corresponding angles are equal/congruent. (Both interior and exterior angles are the same)
  • The ratio of the corresponding sides is the same for all sides. Hence, the perimeters are different.

Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same (the fourth angles automatically become the same as the interior angle sum is 360 degrees), and two adjacent sides have equal ratios.

Difference Between Similarity and Congruence

The words similarity and congruence are associated with shape and size in geometry. Congruence means the same structure, size, and shape whereas similar figures mean the same shape but different size. Let us look at the difference between both terms.

Important Notes

  • If two angles of two triangles are equal then their third angle is always equal.
  • The angle bisector of a triangle always divides the triangle into two similar triangles. (Angle Bisector Theorem)
  • If two similar triangles have sides in ratio \(\frac{x}{y}\) then the ratio of their areas will be \(\frac{x^2}{y^2}\)

Related Topics

Listed below are a few topics related to similar figures, take a look.

  • Dilation Geometry
  • Triangle Congruence Theorem
  • Same Side Interior Angles

Similar Figures Examples

Example 1: Consider the following figure:

Similar Figures Example

Find the value of ∠E.

Match the longest side with the longest side and the shortest side with the shortest side and check all three ratios. We note that the three sides of the two triangles are respectively proportional:

\[\begin{align}& \left\{ \begin{gathered}\frac{{DE}}{{AB}} = \frac{{4.2}}{6} = 0.7\\ \frac{{DF}}{{AC}} = \frac{{2.8}}{4} = 0.7\\ \frac{{EF}}{{BC}} = \frac{{3.5}}{5} = 0.7 \end{gathered} \right.\\&\quad\frac{{DE}}{{AB}} = \frac{{DF}}{{AC}} = \frac{{EF}}{{BC}} \end{align}\]

Thus, by SAS similarity criterion, ΔABC ~ ΔDEF

This means that they are also equiangular. Note carefully that the equal angles will be:

∠A = ∠D = 55.77°

∠C = ∠F = 82.82°

∠E = ∠B = 180° - (55.77° + 82.82°)

∠E = 41.41°

Therefore, ∠E = 41.41°.

Example 2: Consider two similar triangles, ΔABC and ΔDEF:

Similar Figures Example

AP and DQ are medians in the two triangles respectively. Show that

\[\frac{{AP}}{{BC}} = \frac{{DQ}}{{EF}}\]

Since the two triangles are similar, they are equiangular. This means that, ∠B = ∠E

\[\begin{align} \frac{{AB}}{{DE}} &= \frac{{BC}}{{EF}}\\ \Rightarrow \quad\frac{{AB}}{{DE}} &= \frac{{BC/2}}{{EF/2}} = \frac{{BP}}{{EQ}} \end{align}\]

Hence, by the SAS similarity criterion, ΔABP ~ ΔDEQ

Thus, the sides of these two triangles will be respectively proportional, and so:

\[\begin{align} \frac{{AB}}{{DE}} &= \frac{{AP}}{{DQ}}\\ \Rightarrow \quad\frac{{AP}}{{DQ}} &= \frac{{BC}}{{EF}}\\ \Rightarrow \quad\frac{{AP}}{{BC}} &= \frac{{DQ}}{{EF}} \end{align}\]

Therefore, \[\frac{{AP}}{{BC}} = \frac{{DQ}}{{EF}}\].

Example 3: Ben has four squares with the following side lengths: Square A, side = 7 inches, Square B, side = 9 inches, Square C, side = 9 inches, Square D, side = 8 inches. He wants two squares that can be placed exactly one over the other. Can you help him choose the congruent squares?

Solution: Squares with the same sides will superimpose on each other because they will be congruent. So, Ben should find two squares whose side lengths are exactly the same. In the given list, we can see that Square B and Square C have sides of the same length, that is, 9 inches. Therefore, Ben can choose Square B and C because they can be placed exactly one over the other.

go to slide go to slide go to slide

solving mathematical problems involving two dimensional figures

Book a Free Trial Class

Practice Questions on Similar Figures

Faqs on similar figures, how is similarity used in real life.

The similarity is used in designing, solving problems involving height and distance, etc.

What are the Rules of Similarity?

The three rules of similarity are SSS similarity, SAS similarity, and AA or AAA similarity.

Is SSA a Similarity Theorem?

No, SSA is not a similarity theorem.

What is a Similarity Statement?

When two or more objects or figures appear the same or equal due to their shape, this property is known as a similarity.

What is a SSS Similarity Theorem?

The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar.

What is a SAS Similarity Theorem?

The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.

What is a AA Similarity Theorem?

The AA criterion for triangle similarity states that if the three angles of one triangle are respectively equal to the three angles of the other, then the two triangles will be similar.

What are Similar Polygons?

Two polygons are similar when the corresponding angles are equal/congruent, and the corresponding sides are in the same proportion.

The website is not compatible for the version of the browser you are using. Not all the functionality may be available. Please upgrade your browser to the latest version.

  • Social Media
  • Access Points
  • Print/Export Standards
  • Computer Science Standards
  • Coding Scheme
  • B.E.S.T. Standards
  • Standards Viewer App
  • Course Descriptions
  • Graduation Requirements
  • Course Reports
  • Gifted Coursework
  • Career and Technical Education (CTE) Programs
  • Browse/Search
  • Original Student Tutorials
  • MEAs - STEM Lessons
  • Perspectives STEM Videos
  • STEM Reading Resources
  • Math Formative Assessments
  • Our Review Process
  • Professional Development Programs
  • iCPALMS Tools
  • Resource Development Programs
  • Partnership Programs
  • User Testimonials
  • iCPALMS Account

CPALMS Logo

  • Account Information Profile and Notification Settings
  • Administration Manage Site Features
  • Recycle Bin Delete Items From Your Account
  • Not a member yet? SIGN UP
  • Account Information
  • Administration
  • Recycle Bin
  • Home of CPALMS
  • Standards Info. & Resources
  • Course Descriptions & Directory
  • Resources Vetted by Peers & Experts
  • PD Programs Self-paced Training
  • About CPALMS Initiatives & Partnerships
  • iCPALMS Florida's Platform

MA.912.GR.4.4

Export to Word

7 Formative Assessments

5 Lesson Plans

2 Perspectives Video: Professional/Enthusiasts

2 Perspectives Video: Teaching Ideas

1 Perspectives Video: Expert

  • MFAS Formative Assessments 7

Student Resources

Clarifications

Benchmark instructional guide, connecting benchmarks/horizontal alignment.

  • MA.912.GR.1.6 
  • MA.912.GR.3.4 
  • MA.912.T.1.2

Terms from the K-12 Glossary

Vertical alignment.

Previous Benchmarks MA.5.GR.2  MA.6.GR.2  MA.7.GR.1.1 MA.7.GR.1.2  MA.8.GR.1.2  MA.912.AR.2.1 Next Benchmarks MA.912.C.5

Purpose and Instructional Strategies

  • Instruction includes reviewing units and conversions within and across different measurement systems (as this was done in middle grades). 
  • Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form or in terms of pi) or with approximations (e.g., rounding to the 22 nearest tenth or hundredth or using 3.14, 22 7 or other approximations for pi). It is also 7 important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts. 
  • Instruction includes exploring the area of regular polygons and the formula based on the perimeter and the apothem ( A = 1 2 a p , where a is the length of the apothem and p is the perimeter). The apothem is the line segment from the center to the midpoint of on the sides of a regular polygon. In many cases, finding the length of the apothem will require the use of trigonometric ratios. 
  • The population density based on area is calculated by the quotient of the total population and the total area. Have students practice finding the population density or the total population, given the dimensions of a two-dimensional figure. That is, part of their work includes finding the area based on the dimensions. ( MTR.7.1 ) 
  • Instruction includes exploring a variety of real-world situations where finding the area is relevant for different purposes. Problem types include components like percentages, cost and budget, constraints, comparisons and others. 
  • Problem types include finding missing dimensions given the area of a two-dimensional figure or finding the area of composite figures.

Common Misconceptions or Errors

  •  For example, since there are 100 centimeters in a meter, a student may incorrectly conclude that there are 100 square centimeters in a square meter.

Instructional Tasks

  •  Part A. Which county has a higher population density? 
  •  Part B. If the physical shape of the county identified in Part A was a rectangle, what are possible dimensions of the county if the length is greater than the width? 
  •  Part C. If the county identified in Part A was the physical shape of a right triangle, what are possible dimensions of the base and height of the county? 
  •  Part D. Does changing the shape of the tract of land change the population density of the county? 
  • The area of a regular decagon is 24.3 square meters. Determine the side length, in meters, of the regular decagon.

Instructional Items

  • In 2019 the population for Siesta Key, FL, was 5,573 while Destin, FL, had a population of 13,702. Siesta Key is 3.475 square miles and Destin is 8.46 square miles. Which location has a smaller population density?

Related Courses

Related access points, related resources, formative assessments.

This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n -gon inscribed in a circle. They are asked to use the formula for the area of the n -gon to derive an equation that describes the relationship between the area and circumference of the circle.

Type: Formative Assessment

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A (1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r , A ( r ), can be found in terms of the area of the unit circle, A (1) [i.e.,  A ( r ) = r 2 · A (1)].

Lesson Plans

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

Students will construct the medians of a triangle then investigate the intersections of the medians.

Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.

Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

Perspectives Video: Expert

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Download the CPALMS Perspectives video student note taking guide .

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiasts

Go behind the scenes and learn about science museum exhibits, design constraints, and engineering workflow! Produced with funding from the Florida Division of Cultural Affairs.

Type: Perspectives Video: Professional/Enthusiast

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

Perspectives Video: Teaching Ideas

Dr. David McNutt explains how a simple do-it-yourself quadrat and a transect can be used for ecological sampling to estimate population density in a given area.

Type: Perspectives Video: Teaching Idea

Set sail with this math teacher as he explains how kites were used for lessons in the classroom.

Related Resources: KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX] KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

MFAS Formative Assessments

Student resources, perspectives video: professional/enthusiast, parent resources.

solving mathematical problems involving two dimensional figures

Like us on Facebook

Stay in touch with CPALMS

solving mathematical problems involving two dimensional figures

Follow Us on Twitter

CPALMS Logo

Loading....

solving mathematical problems involving two dimensional figures

  • Two-dimensional figures

Grade 5 Philippines School Math Two-dimensional figures

Student assignments, create unlimited student assignments., online practice, online tests, printable worksheets and tests.

IMAGES

  1. 4th Grade Two Dimensional Figures Task Cards

    solving mathematical problems involving two dimensional figures

  2. Lesson 29: Classifying Two Dimensional Figures (Sessions 1 & 2)

    solving mathematical problems involving two dimensional figures

  3. Math

    solving mathematical problems involving two dimensional figures

  4. Grade 1 Math 12.5, Problem Solving, Make new two-dimensional shapes

    solving mathematical problems involving two dimensional figures

  5. (4.G.2) 2-Dimensional Figures: 4th Grade Common Core Math Worksheets

    solving mathematical problems involving two dimensional figures

  6. Space 2D Problem Solving

    solving mathematical problems involving two dimensional figures

VIDEO

  1. Geometry 1-6 Two-Dimensional Figures

  2. Chapter 4 One Dimensional Problems

  3. Dimensional Analysis

  4. Three Dimentional Geometry- Section Formula

  5. Reveal math integrated 1

  6. Math Problem

COMMENTS

  1. Math Problem-Solving Week 7: 2-Dimensional Geometric Shapes

    Day 1 For this problem, I give each student a blackline master of a circle and some colored square tiles. I challenge them to fill the circle completely with the square tiles. They cannot overlap the tiles and they must cover the circle completely--no white spaces left.

  2. Lesson 10

    Lessons 10 and 11 engage students in finding area and circumference/perimeter measurements of different geometric shapes including circles, quadrilaterals, and triangles. The Problem Set Guidance includes a variety of problems that range in difficulty and time required. Students engage in MP.1 and MP.7 as they make sense of complex geometric ...

  3. 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.

  4. MA.912.GR.1.6

    4 Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures. Clarifications Clarification 1: Instruction includes demonstrating that two-dimensional figures are congruent or similar based on given information. General Information Subject Area: Mathematics (B.E.S.T.) Grade: 912

  5. Standards Mapping

    7.G.A.1 Fully covered Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Construct scale drawings Corresponding points and sides of scaled shapes Corresponding sides and points Explore scale copies

  6. 2-Dimensional Shapes and Their Properties

    If students understand the difference between 2-dimensional and 3-dimensional shapes, they should be able to recognize that the cylinder and rectangular prism have height in addition to length and width, and are not flat, hence they're not 2-dimensional objects. Properties of 2-Dimensional Figures

  7. Two-dimensional figures

    Grade 7 (Virginia) 9 units · 103 skills. Unit 1 Rational numbers. Unit 2 Expressions & equations. Unit 3 Inequalities. Unit 4 Ratios & proportions. Unit 5 Transformations & graphing relationships. Unit 6 Two-dimensional figures. Unit 7 Three-dimensional figures. Unit 8 Exponents & scientific notation.

  8. MA.6.GR.2

    4 64 General Information Number: MA.6.GR.2 Title: Model and solve problems involving two-dimensional figures and three-dimensional figures. Type: Standard Subject: Mathematics (B.E.S.T.) Grade: 6 Strand: Geometric Reasoning

  9. 7th Grade Math

    7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Search 7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship ...

  10. (MA.7.GR.1) Solve problems involving two-dimensional figures ...

    Teaching resources aligned to the Mathematics CPALMS for the seventh grade classroom. Including presentations, worksheet printables, projects, interactive activities, assessments, and homework materials that help teach children to solve problems involving two-dimensional figures, including circles.

  11. MA.7.GR.1

    41 13 3 General Information Number: MA.7.GR.1 Title: Solve problems involving two-dimensional figures, including circles. Type: Standard Subject: Mathematics (B.E.S.T.) Grade: 7 Strand: Geometric Reasoning

  12. (MA.6.GR.2) Model and solve problems involving two-dimensional figures

    Calculating Area Activity. Teaching resources aligned to the Mathematics CPALMS for the sixth grade classroom. Including presentations, worksheet printables, projects, interactive activities, assessments, and homework materials that help teach children to model and solve problems involving two-dimensional figures and three-dimensional figures.

  13. PDF Mathematics/Grade7 Unit 5: Two and Three Dimensional Geometry

    number sense computation with whole numbers and decimals, including application of order of operations addition and subtraction of common fractions with like denominators measuring length and finding perimeter and area of rectangles and squares characteristics of 2-D and 3-D shapes angle measurement

  14. Grade 1 Math 12.5, Problem Solving, Make new two-dimensional shapes

    Solving a word problem that involves making a new shape from given shapes, manipulating two-dimensional shapes to create and copy another shape. Working with...

  15. GED Math Practice Questions: Working with Two-Dimensional Figures

    The GED Math test will pose questions where you have to find the perimeter, circumference, or area of a shape, using the appropriate formula. These problems may look very different from each other, but they all involve two-dimensional shapes. The following practice questions involve using measurements in different formulas to get your answers.

  16. PDF Oregon Math Standards

    3. Solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and 4. 7.NS Drawing inferences about populations based on samples. Link to summary of Grade 7 Critical Areas • Students should spend the large majority 1

  17. Lesson 16

    Geometry Lesson 16 Math Unit 6 7th Grade Lesson 16 of 21 Objective Identify and describe two-dimensional figures that result from slicing three-dimensional figures. Common Core Standards Core Standards

  18. Similar figures

    Similar Figures Definition. When two or more objects or figures appear the same or equal due to their shape, this property is known as a similarity or similar figures. When we magnify or demagnify these figures, they always superimpose each other. In geometry, when two shapes such as triangles, polygons, quadrilaterals, etc have the same ...

  19. MA.912.GR.4.4

    Solve mathematical and real-world problems involving the area of two-dimensional figures. Examples Example : A town has 23 city blocks, each of which has dimensions of 1 quarter mile by 1 quarter mile, and there are 4500 people in the town.

  20. Jessa Vill Lopez

    Student Learning Outcomes; Students can solve mathematical problems involving two-dimensional figures. Identify and classify two to three dimensional shapes. Use concrete objects or modeling to solve addition and subtraction problems Draw two and three dimensional shapes (square, triangle and rectangle)

  21. (MA.6.GR.2.3) Solve mathematical and real-world problems involving the

    Teaching resources aligned to the Mathematics CPALMS for the sixth grade classroom. Including presentations, worksheet printables, projects, interactive activities, assessments, and homework materials that help teach children to solve mathematical and real-world problems involving the volume of right rectangular prisms with positive rational number edge lengths using a visual model and a formula.

  22. Grade 5 Philippines School Math Two-dimensional figures

    Two-dimensional figures problems, practice, tests, worksheets, questions, quizzes, teacher assignments | Grade 5 | Philippines School Math

  23. activities that involves mathematical problem solving using two

    A plane figure or two-dimensional figure is a figure that lies completely in one plane. When you draw, either by hand or with a computer program, you draw two-dimensional figures. Blueprints are two-dimensional models of real-life objects. ... These figures are polygons. These figures are not polygons.