Similar Triangles and Proportions: Find Missing Sides and Measure Heights
Use proportions to find missing sides of similar triangles, measure the height of trees and buildings with shadows, and understand how area and volume scale.
Two shapes are similar when one is an enlargement or reduction of the other: the same angles, with every side scaled by the same factor. For triangles this gives a powerful tool, because matching sides form a proportion, and a proportion with one unknown can always be solved.
When are triangles similar?
Any one of these is enough to prove two triangles are similar:
- AA: two pairs of matching angles are equal (the third pair then must be too).
- SSS: all three pairs of matching sides are in the same ratio.
- SAS: two pairs of sides are in the same ratio and the angles between them are equal.
Finding a missing side
Triangle ABC is similar to triangle DEF, with AB = 6, BC = 9, and DE = 10. Find EF.
AB / DE = BC / EF → 6 / 10 = 9 / EF
Cross multiply: 6 × EF = 90, so EF = 15. The scale factor from the smaller triangle to the larger is 10 ÷ 6 ≈ 1.67, and 9 × 1.67 = 15 confirms it. Enter 6, 10, 9 and leave the last box empty in the proportion calculator to see the working.
The key is matching sides correctly: the side opposite a given angle in one triangle matches the side opposite the equal angle in the other. Labelling the vertices in the same order (ABC ~ DEF) keeps this straight.
Measuring a tree with its shadow
On a sunny day, the sun's rays hit a tree and a person at the same angle, so the triangles formed by each object and its shadow are similar.
- A person 5.5 ft tall casts a 4 ft shadow.
- The tree's shadow is 30 ft long.
5.5 / 4 = h / 30 → 4h = 165 → h = 41.25 ft
This method, used since ancient times, works for flagpoles and buildings too. Measure both shadows at the same moment, on level ground.
Scale factors for area and volume
If the sides of similar shapes are in the ratio k, their areas are in the ratio k² and their volumes in the ratio k³. Doubling every side of a box multiplies its surface area by 4 and its volume by 8. That's why a pizza 16 inches across has four times the area of an 8-inch pizza, not twice.
| Length ratio | Area ratio | Volume ratio |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 0.5 | 0.25 | 0.125 |
Real-world uses
- Scale drawings and maps: every feature is similar to the real one; see map scales.
- Photography: image size on a sensor is proportional to object size divided by distance.
- Roof pitch and ramps: a 4-in-12 pitch rises 4 units for every 12 across, the same shape at any size.
- Surveying: distances across rivers can be found from similar triangles laid out on one bank.
Common mistakes
- Pairing sides that aren't corresponding.
- Using the length ratio for area or volume problems.
- Measuring shadows at different times of day.
Further reading from official sources
- Proportional relationships (Grade 7 math) – Khan Academy
