About the Similar Triangles Calculator
Two triangles are similar when they have the same shape — equal angles and proportional sides — even if they are different sizes. This similar triangles calculator has two modes. In “find missing sides” mode, enter all three sides of the first triangle and the side of the second triangle that corresponds to side a; it works out the scale factor and the other two sides. In “check similarity” mode, enter all six sides and it tells you whether the triangles are similar by the SSS rule.
It is useful for geometry homework, for indirect measurement (finding the height of a tree or building from its shadow), for map and model scale problems, and for checking that an enlarged or reduced drawing kept its proportions.
Both modes also report each triangle’s area using Heron’s formula, and the area ratio, which is always the square of the scale factor.
With the default inputs, the scale factor (triangle 2 ÷ triangle 1) is 3. Change any value above to recalculate instantly.
How to use the similar triangles calculator
- 1Choose “find missing sides” or “check if similar”.
- 2Enter the three sides of the first triangle.
- 3Enter the side of the second triangle that matches side a (and, in check mode, the other two).
- 4Read the scale factor, missing sides and the area ratio.
Formula and method
In similar triangles every pair of corresponding sides has the same ratio, called the scale factor k. Once you know one pair of corresponding sides, k = a₂/a₁, and each unknown side of the second triangle is k times its partner in the first. Perimeters scale by k and areas by k², because area is a product of two lengths.
In check mode the sides of each triangle are sorted from shortest to longest (the shortest side is always opposite the smallest angle, so sorted sides correspond). The triangles are similar by SSS when all three ratios are equal, within a 0.01% rounding tolerance. Areas come from Heron’s formula, √(s(s − a)(s − b)(s − c)) with s the semi-perimeter.
- a₁, b₁, c₁
- Sides of the first triangle
- a₂, b₂, c₂
- Corresponding sides of the second triangle
- k
- Scale factor
Worked examples
Scale a 3-4-5 triangle so side a becomes 9
The scale factor is 9 ÷ 3 = 3, so the other sides become 4 × 3 = 12 and 5 × 3 = 15. The area grows by 3² = 9, from 6 to 54.
Are 6-8-10 and 9-12-15 similar?
All three side ratios are 9/6 = 12/8 = 15/10 = 1.5, so the triangles are similar by SSS. The area ratio is 1.5² = 2.25 (24 → 54).
Tree height from its shadow
A 2 m pole casts a 1.5 m shadow while a tree casts a 12 m shadow. The sun’s rays make similar triangles, so the scale factor is 12 ÷ 1.5 = 8 and the tree is 2 × 8 = 16 m tall.
Frequently asked questions
How do you know if two triangles are similar?+
Triangles are similar if two angles match (AA), if all three side ratios are equal (SSS), or if two side ratios are equal and the included angles match (SAS). This tool checks the SSS rule from side lengths.
How do I find a missing side of similar triangles?+
Divide a known side of the second triangle by the matching side of the first to get the scale factor, then multiply any other side of the first triangle by it. For 3-4-5 scaled so 3 becomes 9, k = 3 and the sides are 12 and 15.
How does area change with the scale factor?+
Area scales with the square of the scale factor. Doubling every side (k = 2) makes the area four times larger; halving the sides makes the area one quarter.
Are congruent triangles similar?+
Yes. Congruent triangles are similar with a scale factor of exactly 1 — same shape and same size.
How does the shadow method for measuring height work?+
At the same moment, the sun hits a vertical pole and a tall object at the same angle, forming similar right triangles. Height ÷ shadow is the same for both, so object height = pole height × object shadow ÷ pole shadow.