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.

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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: Indirect measurement finds an unreachable length by setting up a proportion from similar figures.

Common stuck point: The procedure for indirect measurement is the easy part; the trap is pairing the wrong sides in the proportion. Asking "Am I finding an unreachable length by matching corresponding sides of similar figures in a proportion?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I finding an unreachable length by matching corresponding sides of similar figures in a proportion?

Worked Examples

Example 1

medium
A tree casts a shadow of 1515 m at the same time a 22 m pole casts a shadow of 33 m. How tall is the tree?

Answer

The tree is 1010 m tall.

First step

1
Set up a proportion using similar triangles: tree heighttree shadow=pole heightpole shadow\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{pole height}}{\text{pole shadow}}.

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Example 2

hard
From a point 6060 m from the base of a building, the angle of elevation to the top is 50°50°. Find the height of the building to the nearest metre.

Example 3

medium
From 80 m away the angle of elevation to a tower top is 35°. Find the tower height. (Use tan⁡35°≈0.700\tan 35°\approx 0.700.)

Example 4

medium
A 2 m vertical stick casts a 3 m shadow. A nearby pole's shadow is 18 m. Find the pole's height.

Example 5

hard
From the base of a hill, the angle of elevation to a flag is 30°. Walking 50 m closer, the angle is 45°. Find the height of the flag.

Example 6

challenge
A surveyor stands at point PP and sights two points AA and BB across a chasm with ∠APB=40°\angle APB=40°. The surveyor measures PA=120PA=120 m, PB=200PB=200 m. Find ABAB using the Law of Cosines.

Practice Problems

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

Example 1

easy
A 66-foot person casts a 44-foot shadow. At the same time, a nearby flagpole casts a 2020-foot shadow. How tall is the flagpole?

Example 2

medium
To measure river width, a surveyor marks AA and BB on one bank with AB=80AB = 80 m. A tree TT on the other bank is directly opposite AA (so ∠TAB=90°\angle TAB = 90°). From BB, the angle ∠TBA=52°\angle TBA = 52°. Find width ATAT.

Example 3

easy
A 2 m stick casts a 1.6 m shadow. A flagpole casts an 8 m shadow at the same time. How tall is the flagpole?

Example 4

easy
A 10 ft pole and a tree are casting shadows. The pole's shadow is 8 ft; the tree's shadow is 32 ft. Find the tree's height.

Example 5

easy
On a map with scale 1:50,000, two cities are 8 cm apart. What is the real distance?

Example 6

easy
True or false: indirect measurement requires similar triangles.

Example 7

medium
A flagpole's shadow length is 12 m, and the sun's altitude is 60°. Find the flagpole's height. (Use tan⁡60°≈1.732\tan 60°\approx 1.732.)

Example 8

medium
To measure a pond's width, two stakes are set on one shore with AB=50AB=50 m. From BB, a tree across the water makes angle ∠TBA=70°\angle TBA=70° with ABAB; the tree sits directly across from AA. Find the pond width. (Use tan⁡70°≈2.747\tan 70°\approx 2.747.)

Example 9

medium
On a photo, a 2 m tall person measures 4 cm. A pyramid measures 36 cm. Estimate the pyramid's height.

Example 10

medium
A scout sights a cliff using a clinometer reading 38° from 100 m away. Find the cliff height. (Use tan⁡38°≈0.781\tan 38°\approx 0.781.)

Example 11

medium
A 6-ft tall student is sighting a tree top. Holding a ruler 24 inches from the eye, the tree top aligns with the 8-inch mark and the tree's base with the 0 mark. The student is 40 ft from the tree. Find the tree's total height.

Example 12

medium
Why do we verify that two triangles are similar before using a proportion?

Example 13

hard
A 1 m stick casts a 0.5 m shadow. A nearby building's shadow extends 8 m across the ground and then 2 m up an opposite wall. Find the building's height.

Example 14

hard
To measure a tree across a river, a surveyor uses two stakes 60 m apart on the near shore. From the first stake the tree subtends an angle of 30° with the shoreline; from the second stake (closer along the shore), it subtends 90°. Find the perpendicular distance from the second stake to the tree.

Example 15

hard
A radar sends a pulse to an aircraft; the round trip takes 200μs200\mu s. Given the speed of light 3×1083\times 10^8 m/s, find the distance to the aircraft.

Example 16

challenge
Astronomers measure stellar parallax: a star shifts by 0.5 arcseconds when Earth moves through a baseline of 2 AU (= 3×10113\times 10^{11} m). Estimate the distance to the star in meters. (Use 11 arcsecond =4.85×10−6=4.85\times 10^{-6} rad.)
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Background Knowledge

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

similarityproportional geometrysimilarity criteria