Similar Triangles Calculator

Find all corresponding sides and angles of similar triangles given the sides of one triangle and one known value of the second.

Enter Triangle Dimensions

Triangle 1 (all three sides)

Triangle 2 (one known side or scale factor)

Result

Scale Factor (k)
2
ratio
Side a'6
Side b'8
Side c'10
Angle A36.87°
Angle B53.13°
Angle C90°

Step-by-Step Solution

a'/a = b'/b = c'/c = k

Understanding Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This means one triangle is a scaled version of the other. The ratio of any pair of corresponding sides is called the scale factor (k).

If Triangle 1 has sides a, b, c and Triangle 2 has sides a', b', c', then: a'/a = b'/b = c'/c = k.

Conditions for Similarity

AA (Angle-Angle)

If two angles of one triangle are equal to two angles of another, the triangles are similar.

Angle A = Angle A', Angle B = Angle B'

SSS (Side-Side-Side)

If all three pairs of corresponding sides are proportional, the triangles are similar.

a/a' = b/b' = c/c'

SAS (Side-Angle-Side)

If two pairs of sides are proportional and the included angles are equal, the triangles are similar.

a/a' = b/b', Angle C = Angle C'

How to Find Missing Sides

Once you know the scale factor k, multiply each side of Triangle 1 by k to get the corresponding side of Triangle 2. Since similar triangles have equal corresponding angles, you can find all angles using the Law of Cosines on Triangle 1 and those same angles apply to Triangle 2.

Law of Cosines for Angles

Given sides a, b, c of a triangle, you can find angle A (opposite side a) using:

cos(A) = (b² + c² - a²) / (2bc)

Practical Applications

  • Architecture and engineering: scaling blueprints and models.
  • Surveying: measuring inaccessible distances using proportional triangles.
  • Computer graphics: resizing and transforming shapes.
  • Photography: understanding perspective and focal length relationships.