Applying the Area Ratio Theorem
Simple Explanation
The area ratio theorem also works in reverse: if you're given the ratio of two similar triangles' areas, you can find the ratio of their corresponding sides by taking the SQUARE ROOT of the area ratio.
Why Do We Need It?
Exam and design problems just as often hand you the area ratio and ask for the side ratio as the other way around — recognizing when to square and when to take a square root is the real skill.
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 the side ratio from a given area ratio
Two similar triangles have areas 50 and 98. Find the ratio of their corresponding sides, in simplest form.
Why Does This Work?
Since Area₁/Area₂ = (s₁/s₂)², solving for the side ratio just means undoing the squaring — take the square root of both sides of the equation, giving s₁/s₂ = √(Area₁/Area₂). This is valid because all lengths and areas here are positive.
Real-Life Example
Scaling a boat sail to match a target wind-catching area
A sailmaker knows the wind-catching area a new, larger triangular sail needs relative to an existing one, and must find how much bigger to make each edge.
Taking the square root of the target area ratio gives the exact linear scale factor to enlarge every edge of the sail pattern by.
Practice
Two similar triangles have areas 36 and 121. Find the ratio of their corresponding sides, expressed as the smaller side's number in a ratio out of 11 (i.e. find the numerator, if the ratio is written as x : 11).
MediumCommon mistake
Forgetting to take the square root when going from an area ratio to a side ratio — using the area ratio itself as if it were the side ratio.
Quick Review
- s₁/s₂ = √(Area₁/Area₂) — the reverse direction of the area ratio theorem.
- Given areas, simplify the fraction first, then take the square root.
- Squaring goes side ratio → area ratio; square-rooting goes area ratio → side ratio.