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Similar Triangles Calculator

Find the missing sides of a similar triangle from one matching side, scale a triangle by a factor, or check whether two triangles are similar.

Level: Grades 9–10

Printable worksheet

Similar Triangles Calculator

Inputs
What do you want to do?

Result

Triangle 2 sides (a₂, b₂, c₂)

9, 12, 15

With scale factor 3, triangle 2 has sides 9, 12, 15 and 9 times the area of triangle 1.

Scale factor (k)
3
Side a₂
9
Side b₂
12
Side c₂
15
Perimeter of triangle 1
12
Perimeter of triangle 2
36
Area of triangle 1
6 square units
Area of triangle 2
54 square units
Area ratio (k²)
9
Angles (the same in both triangles)
A = 36.8699°, B = 53.1301°, C = 90°

See the step-by-step solution ↓

Step-by-step solution

  1. Matching sides of similar triangles share one ratio. So divide the side you know in triangle 2 by its partner a₁ to get the scale factor k

    k=93=3

  2. Multiply side b₁ by the scale factor k to get its partner b₂

    b₂=3×4=12

  3. Multiply side c₁ by the scale factor k to get its partner c₂

    c₂=3×5=15

  4. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

    Area₂Area₁=32=9

  5. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

    Area₂=6×9=54

  6. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

    9, 12, 15

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find the missing sides of a triangle that has the same shape as one you already know. Two triangles are similar when they have the same shape, but not always the same size. Their angles are equal, and every side of one is the same multiple of the matching side of the other. That multiple is the scale factor. Once you know it, each missing length takes just one multiplication.

The calculator works three ways. Give the three sides of one triangle and one matching side of the other, and it finds the remaining sides. Give a scale factor, and it enlarges or shrinks a triangle. Or enter two sets of sides to check whether the triangles are similar at all. It also shows the shared angles, both perimeters and both areas. You'll see that area grows with the square of the scale factor.

Where it's used

Why this math matters outside the classroom:

  • Shadow measurement. A hiker 1.8 m tall with a 2.4 m shadow estimates a tree with a 16 m shadow is 16 ÷ 2.4 × 1.8 = 12 m tall.
  • Scale drawings. A drafter enlarging a triangular bracket from a 1:4 drawing multiplies a 3 cm edge by 4 to cut a 12 cm part.
  • Map distances. A surveyor uses similar triangles across a river: if 20 m on the near bank matches 8 m in a small triangle, a 5 m leg means 12.5 m.
  • Photo enlargement. A print shop scaling a triangular logo by 1.5 knows its ink area grows by 1.5² = 2.25 times.
  • Ramp design. A builder copying a ramp that rises 0.5 m over 6 m makes a shorter one rising 0.25 m over 3 m, keeping the same slope.

Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSG.SRT.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what you want to do: find missing sides, scale by a factor, or check similarity.
  2. Enter the three sides of triangle 1, labelled a₁, b₁ and c₁.
  3. For missing sides, choose which side of triangle 2 you know and enter its length. For scaling, enter the factor. For a check, enter triangle 2's three sides in any order.
  4. Read the sides of triangle 2 (or the verdict). Then see the scale factor, areas and angles below.

What the inputs mean

Which side of triangle 2 do you know?
Matching sides sit across from equal angles. In a pair of similar triangles, the shortest side matches the shortest, the longest matches the longest, and so on. So make sure a₂ really goes with a₁.
Scale factor (k)
The scale factor is triangle 2's side divided by triangle 1's matching side. A factor of 2 doubles every side, and 0.5 halves them.

Formula and method

k = a₂ ÷ a₁ = b₂ ÷ b₁ = c₂ ÷ c₁
a₂ = k × a₁,  b₂ = k × b₁,  c₂ = k × c₁
perimeter₂ = k × perimeter₁,  area₂ = k² × area₁
Heron: area = √(s(s − a)(s − b)(s − c)),  s = (a + b + c) ÷ 2

In the formulas, a₁, b₁ and c₁ are the sides of triangle 1, and a₂, b₂ and c₂ are the matching sides of triangle 2. k is the scale factor. In Heron's formula, a, b and c are one triangle's sides and s is half its perimeter. Similarity means one triangle is an exact enlargement or shrinking of the other. So every length is multiplied by the same factor k. Divide any side of triangle 2 by its partner in triangle 1, and you get k. Multiply the other sides of triangle 1 by k, and you get the rest of triangle 2.

Perimeter is a sum of lengths, so it grows by k too. Area is a length times a length (a base times a height). So it grows by k × k = k². Doubling the sides of a triangle makes its area four times as large, not twice.

To check similarity from three sides (the side-side-side or SSS test), sort each triangle's sides from shortest to longest. Then divide matching pairs. If all three ratios are equal, the triangles are similar. If any differs, they aren't. The areas here come from Heron's formula. The angles come from the law of cosines, a rule linking a triangle's sides to its angles, and they're the same in both triangles.

Worked example

A person 1.8 m tall casts a 2.4 m shadow at the same moment a tree casts a 16 m shadow. The sun's rays make two similar right triangles. How tall is the tree?

  1. Triangle 1 is the person: height a₁ = 1.8 and shadow b₁ = 2.4. The sun's ray runs along the longest side (the hypotenuse), c₁ = 3, from 1.8² + 2.4² = 9.
  2. The tree's shadow matches the person's shadow, so b₂ = 16 and k = 16 ÷ 2.4 = 6.6667.
  3. Tree height: a₂ = 6.6667 × 1.8 = 12 m.
  4. Ray length: c₂ = 6.6667 × 3 = 20 m.
  5. Area ratio: k² = 44.4444, so the tree's triangle has 96 square meters against the person's 2.16.

Result: The tree is 12 m tall. Triangle 2's sides are 12, 16 and 20, and both triangles have angles of about 36.8699°, 53.1301° and 90°.

Understanding the result

A scale factor above 1 means triangle 2 is the larger one. Below 1, it's smaller. The angles never change with scaling. That's why similar triangles are so handy for measuring things you can't reach. An angle you measure on a small triangle is the same angle on the large one.

In check mode, ratios that are close but not equal (for example 2, 2 and 2.01) mean the triangles aren't exactly similar. With measured lengths, small differences can be measuring error. So decide how close is close enough for your purpose.

Assumptions and limitations

  • The three sides of each triangle can form a real triangle. That means the two shorter sides add up to more than the longest.
  • In missing-side mode, the side you enter for triangle 2 really matches the side you chose in triangle 1.
  • All lengths use the same unit.
  • Similarity is checked by the side-side-side (SSS) test only. You might know angles instead (AA), or two sides and the angle between them (SAS). Then solve each triangle first with the triangle solver.
  • Ratios are compared to about nine significant figures (digits that count). Inputs more precise than that are treated as equal.

Frequently asked questions

How do I find the scale factor of similar triangles?

Divide a side of the second triangle by the matching side of the first. Say a 4 cm side matches a 10 cm side. Then the scale factor from the first triangle to the second is 10 ÷ 4 = 2.5.

If the sides double, what happens to the area?

It's multiplied by 2² = 4. Area depends on two lengths multiplied together. So a scale factor of k multiplies area by k². A scale factor of 3 makes the area 9 times as large.

How can I tell whether two triangles are similar?

Any one of three tests is enough. All three pairs of sides are in the same ratio (SSS). Or two pairs of angles are equal (AA). Or two pairs of sides are in the same ratio and the angles between them are equal (SAS). This calculator uses the SSS test.

Are congruent triangles similar?

Yes. Congruent triangles are similar with a scale factor of exactly 1: same shape and same size.

How this calculator is checked

Method: Definition of similar triangles (equal angles, proportional sides), the SSS similarity criterion, area scaling by k², Heron's formula and the law of cosines. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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