CMClearMathAcademy

Similarity Applications

A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

Similarity lets you measure hard-to-reach objects with proportional triangles, shadows, maps, and scale drawings. The key is matching corresponding sides correctly.

What you'll learn

Why it matters: Shadows, maps, and blueprints use similarity to measure tall objects or real distances without direct measurement.

Worked example

Problem. A 6-foot person casts an 8-foot shadow. A tree casts a 20-foot shadow. How tall is the tree?

  1. The triangles are similar because the sun angle is the same.
  2. Set height/shadow = 6/8 = x/20.
  3. x = 15.

Answer: 15 feet

Practice problems

1. A map scale is 1 inch to 5 miles. How many miles are 4 inches?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A map scale is 1 inch to 5 miles. How many miles are 4 inches?
  2. For similarity and scale problems, match corresponding parts and use a constant scale factor or proportion.
  3. 4 x 5 = 20.
  4. Check the result by substituting or estimating: the response should match 20 and make sense in the original problem.

Answer: 20

2. A 3-foot model represents 18 feet. What is the scale factor from model to real?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A 3-foot model represents 18 feet. What is the scale factor from model to real?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 18/3 = 6.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

3. A 5-foot student casts a 7-foot shadow. A flagpole casts a 21-foot shadow. How tall is it?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A 5-foot student casts a 7-foot shadow. A flagpole casts a 21-foot shadow. How tall is it?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 21 is 3 times 7, so height is 3 times 5.
  4. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

4. A drawing is 8 cm wide and the real object is 32 cm wide. What is the scale factor from drawing to real?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A drawing is 8 cm wide and the real object is 32 cm wide. What is the scale factor from drawing to real?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 32/8 = 4.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

5. A 2-inch blueprint length represents 9 feet. What real length does 7 inches represent?

Show solution
  1. Challenge: First identify exactly what the question is asking: A 2-inch blueprint length represents 9 feet. What real length does 7 inches represent?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 9/2 = 4.5 feet per inch.
  4. 7 x 4.5 = 31.5.
  5. Check the result by substituting or estimating: the response should match 31.5 and make sense in the original problem.

Answer: 31.5

Practice this interactively with instant feedback and an AI tutor.

Practice Similarity Applications Take the free placement check

More Geometry lessons