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Similar Polygons and Scale Factors

Polygons with the same shape, but not necessarily the same size.

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Similar Polygons and Scale Factors

What if you were comparing a baseball diamond and a softball diamond? A baseball diamond is a square with 90 foot sides. A softball diamond is a square with 60 foot sides. Are the two diamonds similar? If so, what is the scale factor?

Similar Polygons and Scale Factors 

Similar polygons are two polygons with the same shape, but not necessarily the same size. Similar polygons have corresponding angles that are congruent, and corresponding sides that are proportional.

These polygons are not similar:

Think about similar polygons as enlarging or shrinking the same shape. The symbol is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. If two polygons are similar, we know the lengths of corresponding sides are proportional. In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the scale factor. The ratio of all parts of a polygon (including the perimeters, diagonals, medians, midsegments, altitudes) is the same as the ratio of the sides.

Understanding a Similarty Statement 

Suppose ABCJKL. Based on the similarity statement, which angles are congruent and which sides are proportional?

Just like in a congruence statement, the congruent angles line up within the similarity statement. So, AJ,BK, and CL. Write the sides in a proportion: ABJK=BCKL=ACJL. Note that the proportion could be written in different ways. For example, ABBC=JKKL is also true.

Solving for Unknown Vlaues 

MNPQRSTU. What are the values of x,y and z?

In the similarity statement, MR, so z=115. For x and y, set up proportions.

1830450x=x25=30x=15 1830=15y18y=450y=25

Solving for the Scale Factor and an Unknown Length 

ABCDAMNP. Find the scale factor and the length of BC.

Line up the corresponding sides, AB and AM=CD, so the scale factor is 3045=23 or 32. Because BC is in the bigger rectangle, we will multiply 40 by 32 because 32 is greater than 1. BC=32(40)=60.

Baseball/Softball Diamond Problem Revisited

All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball diamond. This means all the sides are in a 9060=32 ratio. All the angles in a square are congruent, all the angles in both diamonds are congruent. The two squares are similar and the scale factor is 32.


Example 1

ABCD and UVWX are below. Are these two rectangles similar?

All of the corresponding angles are congruent because the shapes are rectangles. 

Let’s see if the sides are proportional. 812=23 and 1824=34. 2334, so the sides are not in the same proportion, and the rectangles are not similar.

Example 2

What is the scale factor of ABC to XYZ? Write the similarity statement.

All the sides are in the same ratio. Pick the two largest (or smallest) sides to find the ratio.


For the similarity statement, line up the proportional sides. ABXY,BCXZ,ACYZ, so ABCYXZ.

Example 3

ABCMNP. The perimeter of ABC is 150, AB=32 and MN=48. Find the perimeter of MNP.

From the similarity statement, AB and MN are corresponding sides. The scale factor is 3248=23 or 32. ABC is the smaller triangle, so the perimeter of MNP is 32(150)=225.


Determine if the following statements are true or false.

  1. All equilateral triangles are similar.
  2. All isosceles triangles are similar.
  3. All rectangles are similar.
  4. All rhombuses are similar.
  5. All squares are similar.
  6. All congruent polygons are similar.
  7. All similar polygons are congruent.
  8. All regular pentagons are similar.
  9. BIGHAT. List the congruent angles and proportions for the sides.
  10. If BI=9 and HA=15, find the scale factor.
  11. If BG=21, find HT.
  12. If AT=45, find IG.
  13. Find the perimeter of BIG and HAT. What is the ratio of the perimeters?

Use the picture to the right to answer questions 14-18.

  1. Find mE and mQ.
  2. ABCDEQLMNP, find the scale factor.
  3. Find BC.
  4. Find CD.
  5. Find NP.

Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.

  1. ABCDEF Solve for x and y.
  2. QUADKENT Find the perimeter of QUAD.
  3. CATDOG Solve for x and y.
  4. PENTAFIVER Solve for x.
  5. MNOXNY Solve for a and b.
  6. Trapezoids HAVEKNOT Solve for x and y.
  7. Two similar octagons have a scale factor of 911. If the perimeter of the smaller octagon is 99 meters, what is the perimeter of the larger octagon?
  8. Two right triangles are similar. The legs of one of the triangles are 5 and 12. The second right triangle has a hypotenuse of length 39. What is the scale factor between the two triangles?
  9. What is the area of the smaller triangle in problem 30? What is the area of the larger triangle in problem 30? What is the ratio of the areas? How does it compare to the ratio of the lengths (or scale factor)? Recall that the area of a triangle is A=12 bh.

Review (Answers)

To view the Review answers, open this PDF file and look for section 7.3. 

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Congruent Congruent figures are identical in size, shape and measure.
Proportion A proportion is an equation that shows two equivalent ratios.
Scale Factor A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.
sine The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.
Trigonometric Ratios Ratios that help us to understand the relationships between sides and angles of right triangles.

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