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.
Showing a random 20 of 50 problems.
Example 1
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 2
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 3
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?Tree triangle: shadow = 15 m, height = h
Example 4
medium
A scout sights a cliff using a clinometer reading 38° from 100 m away. Find the cliff height. (Use tan38°≈0.781.)
Example 5
challenge
A 1 m stick at the edge of a flat field blocks the view of a distant 50 m tower exactly when held 0.4 m from the eye. How far away is the tower?Tower triangle (sighting): height = 50 m, distance x = ?
Example 6
medium
From 80 m away the angle of elevation to a tower top is 35°. Find the tower height. (Use tan35°≈0.700.)
Example 7
easy
What two measurable quantities does the shadow method require?
Example 8
easy
A 5 ft stick casts a 3 ft shadow. A flagpole casts a 24 ft shadow. Find the flagpole's height.Flagpole triangle: shadow = 24 ft, height h = ?
Example 9
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 10
medium
To measure a pond's width, two stakes are set on one shore with AB=50 m. From B, a tree across the water makes angle ∠TBA=70° with AB; the tree sits directly across from A. Find the pond width. (Use tan70°≈2.747.)
Example 11
easy
A photo (scale: 1 cm = 2 m) shows a building 7 cm tall. What is the real height?
Example 12
challenge
From a point, the angle of elevation to a tower top is 30°; moving 40 m closer it becomes 45°. Find the tower height (exact form).
Example 13
challenge
Astronomers measure stellar parallax: a star shifts by 0.5 arcseconds when Earth moves through a baseline of 2 AU (= 3×1011 m). Estimate the distance to the star in meters. (Use 1 arcsecond =4.85×10−6 rad.)
Example 14
easy
Using a mirror placed 3 m from a tree: an observer with eyes 1.5 m above ground stands 1 m from the mirror and sees the tree top. Find the tree height.
Example 15
easy
A 1.5 m tall student casts a 1 m shadow. A nearby building casts a 12 m shadow. Find the building's height.
Example 16
medium
A 1.5 m tall student measures a 30° angle of elevation to a roof from 20 m away. Find the roof's height above the ground (use tan30°≈0.577).
Example 17
easy
On a map with scale 1:50,000, two cities are 8 cm apart. What is the real distance?
Example 18
easy
Why does the shadow method for measuring heights work?
Example 19
medium
A 1.6 m tall student measures an elevation angle of 25° to a treetop from 30 m away. Find the tree height. (Use tan25°≈0.466.)
Example 20
medium
A photo shows a person (real height 1.8 m) as 3 cm tall and a building as 25 cm tall. Estimate the building's real height.