Similar Triangles Calculator (scale factor & missing sides, incl. right triangles)

Understanding the similar triangles calculator — concept and principle

This calculator finds the missing sides of a triangle that is similar to a known one, from the scale factor between them. It checks any extra sides you enter for consistency and gives both areas, the area ratio and the angles. Use it for geometry homework, shadow and height problems, map and model scaling, and indirect measurement.

Every side is multiplied by 2.5, so the area grows by 2.5² = 6.25.

How it is calculated

Scale factor k = (side of B) ÷ (matching side of A), from the first known side of B.
Missing sides of B = k × the matching sides of A. Other known sides of B are checked against k.
Perimeters scale by k; areas by k² (areas from Heron's formula).
Angles, the same in both, come from the law of cosines.

In right-triangle mode, c is the hypotenuse and one side of A may be 0; it is found with a² + b² = c².

The principle

Two triangles are similar when their corresponding angles are equal. Then one is an enlargement or reduction of the other, so every length is multiplied by the same factor k, while angles do not change. Area is length times length, so it is multiplied by k × k = k². Similarity is the basis of indirect measurement: a short stick and a tall tree cast shadows in the same proportion because the sun's rays make the same angles.

Worked example

Default inputs: triangle A with sides 3, 4, 5; triangle B with c = 12.5.

k = 12.5 ÷ 5 = 2.5.
Side a of B = 2.5 × 3 = 7.5; side b of B = 2.5 × 4 = 10.
Area of A (right triangle) = 3 × 4 ÷ 2 = 6.
Area ratio k² = 6.25, so area of B = 6 × 6.25 = 37.5.
Angles in both: 36.87°, 53.13°, 90°.

Each point is the calculator re-run with only “Triangle B: side c (0 = unknown)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Triangle B: side c (0 = unknown)	Scale factor k (B ÷ A)
3.12	0.625
6.25	1.25
9.38	1.875
12.5	2.5
15.6	3.125
18.8	3.75
25	5

How to read the result

k above 1 means B is larger than A; below 1, smaller. If an entered side of B does not match k, the triangles are not similar with that side order, or one measurement is wrong. Remember that areas grow much faster than sides: doubling every side quadruples the area.

Common mistakes

Matching the wrong sides; corresponding sides are opposite equal angles.
Scaling area by k instead of k².
Assuming triangles are similar because they look alike without checking angles or side ratios.
Mixing units between the two triangles.

Keep in mind: Real measurements contain errors, so measured sides rarely give exactly the same ratio. Small differences may be measurement error rather than non-similar shapes.

Frequently asked questions

How do I know two triangles are similar?

By two equal angles (AA), all three side ratios equal (SSS), or two side ratios equal with the included angle equal (SAS).

How do I find the height of a tree with similar triangles?

Compare its shadow with the shadow of an object of known height at the same time: tree height = object height × tree shadow ÷ object shadow.

Why does the area scale by k²?

Area depends on two lengths, and each is multiplied by k.

Sources: OpenStax, Prealgebra 2e (2020), §9.4 (properties of similar triangles)

Checked 9 Nov 2026. Standard similarity theorems of Euclidean geometry (equal corresponding angles ⇔ proportional corresponding sides; areas of similar figures scale with the square of the ratio); Heron’s formula and the law of cosines for area and angles.

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