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. In Triangles, Similarity, and Trigonometry, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

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.
  4. Connect the result back to Similarity Applications so the geometric relationship is explicit.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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