Indirect Measurement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Indirect Measurement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Indirect measurement finds unknown lengths by using proportional relationships instead of direct measuring tools.

Use a smaller, measurable shadow to infer a taller object’s height.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proportions transfer measurements across similar situations.

Common stuck point: Students set up proportions with mismatched corresponding sides.

Sense of Study hint: Label corresponding sides first, then write the proportion in one consistent order.

Worked Examples

Example 1

medium
A tree casts a shadow of 15 m at the same time a 2 m pole casts a shadow of 3 m. How tall is the tree?

Solution

  1. 1
    Set up a proportion using similar triangles: \frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{pole height}}{\text{pole shadow}}.
  2. 2
    Substitute known values: \frac{h}{15} = \frac{2}{3}.
  3. 3
    Cross-multiply: 3h = 30, so h = 10 m.

Answer

The tree is 10 m tall.
When two objects cast shadows at the same time, the sun creates similar triangles. Corresponding sides of similar triangles are proportional, allowing the unknown height to be found from the known height-to-shadow ratio.

Example 2

hard
From a point 60 m from the base of a building, the angle of elevation to the top is 50°. Find the height of the building to the nearest metre.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A 6-foot person casts a 4-foot shadow. At the same time, a nearby flagpole casts a 20-foot shadow. How tall is the flagpole?

Example 2

medium
To measure river width, a surveyor marks A and B on one bank with AB = 80 m. A tree T on the other bank is directly opposite A (so \angle TAB = 90°). From B, the angle \angle TBA = 52°. Find width AT.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

similarityproportional geometrysimilarity criteria