How to Find Missing Sides in Similar Triangles

 

Similar triangles have equal corresponding angles and proportional corresponding sides. Once similarity is established, a known pair of corresponding sides gives a scale factor. You can then apply that factor to another side to find its matching length in the other triangle.

The Similar Triangle Calculator uses two sides from Triangle 1 and one corresponding side from Triangle 2. It calculates the missing side, forward and reverse scale factors, and perimeter and area multipliers. It assumes the triangles are already known to be similar.

Identify corresponding sides

Label two sides of Triangle 1 as a and b. In Triangle 2, let side A correspond to a and side B correspond to b. A rotated or reflected drawing does not change which angles and sides correspond.

Use equal opposite angles or the stated vertex correspondence to match sides. Do not pair them solely because they occupy similar positions on the page. An incorrect pairing can produce a valid-looking proportion with the wrong answer.

Calculate the forward scale factor

The scale factor from Triangle 1 to Triangle 2 is k = A / a. If side a is 3 and corresponding side A is 6, the factor is 6 / 3 = 2.

Every corresponding length in Triangle 2 is twice its matching length in Triangle 1. A factor greater than one indicates enlargement. A positive factor below one indicates reduction, while a factor of one gives equal corresponding lengths.

Find the missing side

Use B = b x k, or equivalently B = b x A / a. If b is 5 and the scale factor is 2, then B = 5 x 2 = 10.

In the calculator, enter Triangle 1 side a 3, Triangle 1 side b 5 and Triangle 2 side A (corresponds to a) 6. Select Calculate missing side. The missing side B is 10 and the forward scale factor is 2.

Keep the scale direction consistent

The reverse scale factor is a / A, or 1 / k. In the example it is 0.5. Multiplying Triangle 2 side B by 0.5 returns Triangle 1 side b: 10 x 0.5 = 5.

Do not use the reverse multiplier when calculating a length in the forward direction. Write down which triangle you start from and which triangle you are finding before forming a ratio.

Work through a reduction

Suppose a = 8, b = 12 and A = 2. The forward scale factor is 2 / 8 = 0.25. Multiply side b by 0.25 to obtain B = 3.

The reverse factor is 4. This checks the relationship because 3 x 4 = 12. All lengths must be positive and use the same unit. Convert centimeters and meters to a common unit before entering them rather than mixing their numeric values.

Compare perimeter and area multipliers

Perimeter scales by k because every side length uses that multiplier. Area scales by k squared because it combines two length dimensions. A scale factor of 2 gives a perimeter multiplier of 2 and an area multiplier of 4.

For the reduction factor 0.25, the area multiplier is 0.25 squared = 0.0625. These are dimensionless relationships. The calculator does not display actual areas or perimeters because the remaining dimensions have not been supplied.

Establish similarity before calculating

Triangle similarity can be established through two equal corresponding angles, three proportional corresponding side pairs, or two proportional side pairs with equal included angles. Use the information provided in the problem to justify the relationship.

Three entries in this calculator do not prove similarity. Nor do they solve an arbitrary triangle or check a drawing's angle conditions. If your problem instead involves a known right triangle, use the Right Triangle Calculator for supported side and angle calculations.

Results are rounded for display. Keep unrounded values when continuing a multi-step calculation, and retain the original measurement units for the missing side.