How to Find the Perimeters of Similar Triangles

By Bill Varoskovic
The perimeter of a triangle is the sum of its three sides.
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In mathematics, like most other areas of life, you are often given a set of facts and then asked to draw conclusions or answer questions. One situation in which you can answer questions from given information involves finding the perimeters of two triangles that are similar. In geometry, a triangle is similar to another triangle if it has the same angles (the same shape) but differs in size. It is possible to find the perimeter of both triangles as long as you have the length of one side of each triangle.

Calculate the side lengths of the first triangle. The Law of Sines says that side a/sine angle A = side b/sine angle B = side c/sine angle C. Using the length of a side that you know, plug in the values of the side (and its corresponding sine) to get a value. Take that value and use it to solve, by cross-multiplication, for length B or C.

For instance, say side a equals 5. The angle opposite it is 40 degrees, and angle B is 20 degrees. The equation would be 5/sine 40 = b/sine 20. Use cross multiplication to determine that b = 2.66.

Add the lengths of each side together to find the perimeter of the first triangle.

Determine the corresponding sides of triangle one and triangle two. The longest sides of each triangle need to match up, as well as the shortest lengths in each pairing.

Calculate a ratio between a length in triangle one and its corresponding side on triangle two, whose length you have been given. For example, if the hypotenuse in triangle one is 8 and in triangle two is 4, the ratio is 1/2.

Multiply this ratio by the perimeter of triangle one to obtain the perimeter of triangle two. For example, if the perimeter of triangle one was 14, multiply 14 by 1/2 to get the perimeter of triangle two. The answer would be 7.