Similar Triangles: How to Convert a Side Ratio into an Area Ratio

How are areas of similar triangles related?

For two similar triangles, the ratio of their areas equals the square of the ratio of corresponding sides. If the side ratio is m:n, the area ratio is m²:n². Conversely, take the positive square root of an area ratio to recover the corresponding side ratio.

Why is the ratio squared?

The area of a triangle is half its base times its perpendicular height. For similar triangles, both corresponding bases and corresponding heights scale by the same factor k. Their product therefore scales by k × k = k². The factor one-half appears in both areas and cancels.

This explanation depends on similarity. Knowing just one pair of sides in unrelated triangles does not determine their area ratio because the heights could be different.

Worked example

Two similar triangles have corresponding sides 6 cm and 9 cm. The small-to-large side ratio is 2:3, so their areas are in the ratio 4:9. If the smaller area is 20 cm², the larger area is 20 × 9/4 = 45 cm².

Check the direction: the larger side must correspond to the larger area. Multiplying by 4/9 instead would contradict that relationship.

Working backwards

Suppose the area ratio is 25:64. The corresponding side ratio is 5:8, not 25:64. If the smaller corresponding side is 15 cm, the larger one is 15 × 8/5 = 24 cm. Perimeters also follow 5:8 because they measure lengths, rather than areas.

Practice with answers

• Side ratio 3:5 gives area ratio 9:25.
• Area ratio 49:81 gives side and perimeter ratio 7:9.
• A triangle is enlarged by a length factor 2.5: its area multiplies by 6.25.

Label the triangles in the same order throughout your calculation. Match corresponding sides using the stated similarity, rather than selecting whichever numbers look convenient.

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