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Similar shapes
Here you will learn about similar shapes, including what they are and how to identify them.
Students will first learn about similar shapes as part of geometry in katex is not definedth grade.
What are similar shapes?
Similar shapes have sides of different lengths, but all corresponding sides are related by the same scale factor. All the corresponding angles in the similar shapes are equal and the corresponding lengths are in the same ratio.
For example, these two rectangles are similar shapes because:
- The scale factor from shape katex is not defined to shape katex is not defined is katex is not defined
- The ratio of the bases is katex is not defined which simplifies to katex is not defined
- The ratio of the heights is also katex is not defined
NOTE: Since the opposite sides of a rectangle are congruent, by comparing the base and the height, you have compared all sides of the rectangle.
- The angles are all katex is not defined
What are similar shapes?
Common Core State Standards
How does this relate to katex is not definedth grade math?
- Grade 8: Geometry (8.G.A.2)
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
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[FREE] 2D Shape Worksheet (Grade 2 to 4)
Use this quiz to check your grade 2 to 4 students’ understanding of 2D shape. 15+ questions with answers covering a range of 2nd, 3rd and 4th grade 2D shape topics to identify areas of strength and support!
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Use this quiz to check your grade 2 to 4 students’ understanding of 2D shape. 15+ questions with answers covering a range of 2nd, 3rd and 4th grade 2D shape topics to identify areas of strength and support!
DOWNLOAD FREEHow to decide if shapes are similar
In order to decide if shapes are similar:
- Decide which sides are pairs of corresponding sides.
- Find the ratios of the sides.
- Check if the ratios are the same.
Similar shapes examples
Example 1: identifying similar rectangles
Are these shapes similar?
- Decide which sides are pairs of corresponding sides.
The bases of the rectangles are a pair of corresponding sides.
The heights of the rectangles are a pair of corresponding sides.
2Find the ratios of the sides.
When writing the ratios, the order is very important.
Here the ratio is katex is not defined.
The ratio of the bases is katex is not defined
The ratio of the heights is katex is not defined which simplifies to katex is not defined
3Check if the ratios are the same.
The ratios for the corresponding lengths are the same.
The scale factor from shape katex is not defined to shape katex is not defined is katex is not defined
The rectangles are similar shapes.
Example 2: identifying similar triangles
Are these shapes similar?
Decide which sides are pairs of corresponding sides.
The bases of the triangles are a pair of corresponding sides.
The heights of the triangles are a pair of corresponding sides.
The right angles are corresponding angles.
NOTE: For right triangles, you can just compare the base and height, without considering the hypotenuse.
Find the ratios of the sides.
When writing the ratios the order is very important.
The ratio of the heights is katex is not defined which simplifies to katex is not defined
The ratio of the bases is katex is not defined This can be written as katex is not defined or katex is not defined
Check if the ratios are the same.
The ratios for the corresponding lengths are NOT the same.
The triangles are NOT similar shapes.
NOTE: This example shows the importance of proving similarity mathematically. Just glancing at these right triangles, they may appear to be similar. However, when you solve for the ratio of their sides, you see that they are not (though they are very close to be similar!)
How to find a missing length
In order to find a missing side in a pair of similar shapes:
- Decide which sides are pairs of corresponding sides.
- Find the scale factor.
- Use the scale factor to find the missing length.
Missing length examples
Example 3: finding a missing length on a quadrilateral
Here are two similar shapes. Find the length katex is not defined
Decide which sides are pairs of corresponding sides.
Identify the corresponding sides that have measurements and those mentioned in the question.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
There are other corresponding sides, but you do not need them to answer the question.
Find the scale factor.
The ratio of the lengths katex is not defined is katex is not defined which simplifies to katex is not defined
This means scale factor from rectangle katex is not defined to katex is not defined is katex is not defined and the scale factor from rectangle katex is not defined to katex is not defined is katex is not defined
Use the scale factor to find the missing length.
The ratio of the lengths katex is not defined is also katex is not defined
Use the scale factor katex is not defined as a multiplier to find the missing length.
The missing side has been found to be katex is not defined
Alternatively, an equation may be formed and solved:
Example 4: finding a missing length
Here are two similar triangles. Find the length katex is not defined
Decide which sides are pairs of corresponding sides.
Identify the corresponding sides that have measurements and those mentioned in the question.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
There are other corresponding sides, but you do not need them to answer the question.
Find the scale factor.
The ratio of the lengths katex is not defined is katex is not defined which simplifies to katex is not defined
This scale factor from triangle katex is not defined to katex is not defined is katex is not defined and the scale factor from triangle katex is not defined to katex is not defined is katex is not defined
But we want a scale factor from katex is not defined to katex is not defined which will be katex is not defined
Use the scale factor to find the missing length.
The ratio of the lengths katex is not defined is also katex is not defined
Use the scale factor katex is not defined as a multiplier to find the missing length.
The missing side is katex is not defined
Alternatively, an equation may be formed and solved:
How to find a missing length in a triangle
In order to find a missing side in a pair of triangles when you are not told that the triangles are similar:
- Use angle facts to determine which angles are equal.
- Redraw the triangles side by side.
- Decide which sides are pairs of corresponding sides.
- Find the scale factor.
- Use the scale factor to find the missing length.
Missing length in a triangle examples
Example 5: finding a missing length in a triangle
Find the value of katex is not defined
Use angle facts to determine which angles are equal.
Look for equal angles.
katex is not defined given that vertically opposite angles are equal.
There are a pair of parallel sides; katex is not defined and katex is not defined This means that there are equal angles since there are alternate angles.
katex is not defined (alternate angles)
katex is not defined (alternate angles)
Redraw the triangles side by side.
Redraw the triangles side-by-side so that it is easier to identify the corresponding sides.
Decide which sides are pairs of corresponding sides.
Identify the corresponding sides that have measurements and those mentioned in the question. Use the angles to help you.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
There are other corresponding sides, but you do not need them to answer the question.
Find the scale factor.
The ratio of the lengths katex is not defined is katex is not defined which simplifies to katex is not defined
This is a scale factor from triangle katex is not defined to katex is not defined of katex is not defined and a scale factor from triangle katex is not defined to katex is not defined of katex is not defined
Use the scale factor to find the missing length.
The ratio of the lengths katex is not defined is also katex is not defined
Use the scale factor katex is not defined as a multiplier to find the missing length.
The value of katex is not defined has been found, katex is not defined
Alternatively, an equation may be formed and solved:
Example 6: finding a missing length in a triangle
Find the value of katex is not defined
Use angle facts to determine which angles are equal.
Look for equal angles.
katex is not defined as they are common to both triangles.
There are a pair of parallel sides – katex is not defined and katex is not defined This means that there are equal angles as there are corresponding angles.
katex is not defined (corresponding angles)
katex is not defined (corresponding angles)
Redraw the triangles side by side.
Redraw the triangles side-by-side so that it is easier to pair up the corresponding sides.
Decide which sides are pairs of corresponding sides.
Identify the corresponding sides that have measurements and those mentioned in the question. Use the angles to help you.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
- The sides katex is not defined and katex is not defined are a pair of corresponding sides.
There are other corresponding sides, but you do not need them to answer the question.
Find the scale factor.
The ratio of the lengths katex is not defined is katex is not defined which simplifies to katex is not defined or katex is not defined
The scale factor from katex is not defined to katex is not defined is katex is not defined.
Use the scale factor to find the missing length.
The ratio of the lengths katex is not defined is also katex is not defined
Use the scale factor katex is not defined as a multiplier to find the missing length.
The value of katex is not defined has been found, katex is not defined
Alternatively, an equation may be formed and solved:
Teaching tips for similar shapes
- Introduce the topic with familiar, regular polygons, before progressing to more complex or irregular shapes.
- Use a variety of activities to teach students this topic, such as worksheets, interactive websites, or manipulatives. You can even challenge students to create similar shapes with given materials, having them justify their conclusions.
Easy mistakes to make
- Confusing the order of ratios
Make sure that you are consistent with your ratios. For example,
In this example, always write the katex is not defined value first, and then the katex is not defined value.
katex is not defined and katex is not defined The ratios are equal, so these shapes are similar shapes.
- Using only what the shapes look like to decide if they are similar
Often diagrams for questions involving similar figures are NOT drawn to scale.
So, use the measurements given, rather than just looking or trying to measure on your own.
- Forgetting that shapes can be similar but in different orientations
The second shape may be in a different orientation to the first shape. The shapes can still be similar. For example, here shape katex is not defined and Shape katex is not defined are similar. Shape katex is not defined is an enlargement of shape katex is not defined by scale factor katex is not defined
Here, shape katex is not defined has been rotated to make the similarity easier to see.
- Thinking scale factor is always a whole number
The scale factor can also be a fraction or a decimal. When the fraction or decimal is less than katex is not defined it means the shape shrank.
Related congruence and similarity lessons
Practice similar shapes questions
1. Consider if these shapes are similar:
Yes – sides in ratio katex is not defined
No – sides in ratio katex is not defined and katex is not defined
No – sides in ratio katex is not defined and katex is not defined
Yes – sides in ratio katex is not defined
The shapes are similar, since the ratios of the corresponding sides are the same.
The ratio of the bases is katex is not defined
which simplifies to katex is not defined
and the ratio of the heights is katex is not defined
2. Consider if these shapes are similar:
Yes – sides in ratio katex is not defined
Yes – sides in ratio katex is not defined
No – sides in ratio katex is not defined and katex is not defined
No – sides in ratio katex is not defined and katex is not defined
Comparing the base and height as shown creates the ratios katex is not defined and katex is not defined which are not equal.
However if you rotate the shapes, they are similar since the ratios of the corresponding sides are the same.
The ratio of the height is katex is not defined
which simplifies to katex is not defined
and the ratio of the base is katex is not defined
which simplifies to katex is not defined
3. These shapes are similar. Find the value of katex is not defined
The ratio of the bases is katex is not defined
which simplifies to katex is not defined
The scale factor from the red parallelogram to the blue is katex is not defined
katex is not defined
4. These shapes are similar. Find the value of katex is not defined
The ratio of the bases is katex is not defined
which simplifies to katex is not defined
or katex is not defined
The scale factor from the small shape to the larger is katex is not defined
katex is not defined
5. Find the value of katex is not defined
Use the parallel lines to identify equal angles.
Then find pairs of corresponding sides.
The ratio of the corresponding sides is katex is not defined which simplifies to katex is not defined
The scale factor from the smaller triangle to the larger is katex is not defined
katex is not defined
6. Find the value of katex is not defined
Use the parallel lines to identify equal angles.
Then find pairs of corresponding sides.
The ratio of the corresponding sides is katex is not defined which simplifies to katex is not defined
The scale factor from the larger triangle to the smaller is katex is not defined
katex is not defined
Similar shapes FAQs
Yes, similar shapes have different side lengths, but the ratio between corresponding side lengths is always equal.
A dilation is when a shape’s size is increased by a scale factor. A dilation creates a similar shape.
See also: Dilations
The theorem states that the sides of a right triangle are related by katex is not defined where katex is not defined and katex is not defined are the sides that form the right angle and katex is not defined is the hypotenuse.
See also: Pythagorean Theorem
The next lessons are
- Transformations
- Mathematical proof
- Area
- Surface area
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