Math
Similar Triangles Calculator
Similar triangles have the same angles and proportional sides.
Enter the three sides of the first triangle and either one corresponding side of the second triangle or the scale factor. The calculator returns the missing sides, the scale factor as a fraction and decimal, and the ratios of perimeters (k) and areas (k²).
The side of the second triangle that matches side a — opposite the same angle.
Try an example
Result
Scale factor k
—
- Triangle 2 sides
- —
- Perimeters (1 → 2)
- —
- Areas (1 → 2)
- —
- Area ratio k²
- —
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Study path
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What the similar triangles calculator solves
Two triangles are similar when their angles match, which forces every pair of corresponding sides to have the same ratio — the scale factor k. Once you know k from one pair of sides, every other side, the perimeter, and the area follow.
| Scale factor k | Sides | Perimeter | Area |
|---|---|---|---|
| 2 | ×2 | ×2 | ×4 |
| 3 | ×3 | ×3 | ×9 |
| 1/2 | ×0.5 | ×0.5 | ×0.25 |
| 2.5 | ×2.5 | ×2.5 | ×6.25 |
How to tell which sides correspond
Corresponding sides sit opposite equal angles. In the similarity statement △ABC ~ △DEF, side AB matches DE, BC matches EF, and CA matches FD. The longest side of one triangle always corresponds to the longest side of the other, which is a quick way to check your pairing.
How to use it
- Enter the three sides of the first triangle.
- Either enter the side of the second triangle that corresponds to side a, or enter the scale factor directly.
- Read k as a fraction and decimal, then the missing sides b′ and c′.
- Use the perimeter ratio (k) and area ratio (k²) for follow-up questions.
How to read the answer
k greater than 1 means triangle 2 is an enlargement; k less than 1 means it is a reduction; k = 1 means the triangles are congruent. The perimeter grows by the same factor as the sides, but the area grows by the square of it.
Common mistakes and edge cases
- Pairing the wrong sides. Match sides that are opposite equal angles, or match longest-to-longest, middle-to-middle, shortest-to-shortest.
- Dividing in the wrong direction. k = triangle 2 ÷ triangle 1 here; flipping it gives 1/k.
- Using k for the area ratio. It is k².
- Sides like 1, 2, 3 form no triangle; the calculator still scales them but warns that there is no area.
Worked examples
3-4-5 scaled so a′ = 7.5
k = 5/2
Scale factor k
5/2 (2.5)
Scale factor 2
6-8-10, area ×4
Scale factor k
2
Reduction: a′ = 1.5
k = 1/2, area ×1/4
Scale factor k
1/2 (0.5)
Congruent: a′ = 3
k = 1
Scale factor k
1
Decimal sides 2.5, 3.5, 4 with k = 1.2
Scale factor k
1.2
Zero side
Scale factor k
—
Frequently asked questions
What is the scale factor of similar triangles?+
The ratio of any pair of corresponding sides: k = side of triangle 2 ÷ matching side of triangle 1. It is the same for all three pairs.
How do I find a missing side in similar triangles?+
Multiply the corresponding known side by the scale factor. If k = 2.5 and side b = 4, then b′ = 10.
Why is the area ratio the square of the scale factor?+
Area is two-dimensional. Scaling both base and height by k multiplies the area by k · k = k².
Can similar triangles be congruent?+
Yes. Congruent triangles are similar with scale factor 1.
How do I know two triangles are similar?+
AA (two equal angles), SSS (all three side ratios equal), or SAS (two side ratios equal and the included angles equal).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026