The Area Ratio Theorem for Similar Triangles
Simple Explanation
If two triangles are similar, the ratio of their areas is not the same as the ratio of their sides — it equals the SQUARE of that ratio. So if every side of a triangle is scaled by a factor k, the area is scaled by k².
Why Do We Need It?
This is a genuinely surprising and useful shortcut: it lets you compare the areas of similar triangles instantly from a single side-length ratio, without recomputing either area from scratch.
See It
A small right triangle with legs 3 and 4 and hypotenuse 5, next to a larger similar triangle with legs 6 and 8 and hypotenuse 10, both shaded to compare area
Formula
Area Ratio of Similar Triangles
Area₁ / Area₂ = (s₁ / s₂)²
For two similar triangles, the ratio of their areas equals the square of the ratio of any pair of corresponding side lengths.
- Area₁, Area₂
- — the areas of the two similar triangles
- s₁, s₂
- — a pair of corresponding side lengths (or the linear scale factor between the two triangles), one from each triangle
When to use it: Whenever you know (or can find) a pair of corresponding side lengths of two similar triangles and need to relate their areas.
Worked Example
Find an unknown area using the area ratio theorem
Two similar triangles have corresponding sides of length 5 and 15. The smaller triangle has area 20. Find the area of the larger triangle.
Why Does This Work?
Stated simply: let the small triangle have base b and height h, so its area is ½bh. Because the triangles are similar, every length scales by the same factor k — including the height, since the height is itself a length inside the triangle's own scaled copy. So the large triangle has base kb and height kh, giving area ½(kb)(kh) = k²(½bh) — exactly k² times the small triangle's area.
Real-Life Example
Estimating mulch for a scaled-up flower bed
A garden designer doubles every side length of a triangular flower bed to make it look grander.
Because area scales with the square of the linear scale factor, doubling every side (k=2) means the bed needs 2²=4 times as much mulch to cover it — not just twice as much, a common and costly mistake.
Practice
Two similar triangles have corresponding sides 6 and 9. The larger triangle has area 81. Find the area of the smaller triangle.
MediumCommon mistake
Applying the side-length ratio directly to the areas (e.g. assuming doubling every side doubles the area) instead of squaring it first — area scaling is always the SQUARE of the linear scale factor.
Quick Review
- Area₁/Area₂ = (s₁/s₂)², for corresponding sides s₁ and s₂.
- Proof idea: both base and height scale by the same factor k, so area (½ × base × height) scales by k².
- Doubling every side (k=2) quadruples the area (k²=4), not doubles it.