How to Find the Perimeter of an Isosceles Triangle Given the Height & Base

Isosceles triangles can have different shapes and sizes.

Isosceles triangles by definition must possess at least two sides that are equal in length; if all three sides are equal, then the triangle is a special type of isosceles called equilateral. For an isosceles triangle with two equal sides, its base is its unequal side and the height is the perpendicular distance from the base to the angle opposite it; and for an equilateral triangle, the base is any one of its sides, with the height maintaining the same properties. The height for either type bisects the triangle, creating two right triangles, which are triangles with one 90-degree angle. By working with the Pythagorean theorem --- which says that the square of the longest leg of a right triangle that is opposite the right angle, called its hypotenuse, is equal to the squares of the other two sides added together --- you can find the perimeter of both types of isosceles triangles.

Things You'll Need

• Calculator

Instructions

1. Halve the length of the base and then square the half. For this example, let the length be 6 inches --- half of 6 inches is 3 inches, and 3 inches squared is 9 square inches.

2. Square the height. In this example, let the height be 4 inches --- the square of 4 inches is 16 square inches.

3. Sum the squares and then find the square root of that sum. In this example, 9 square inches added to 16 square inches equals 25 square inches, and the square root of 25 square inches is 5 inches.

4. Multiply the square root from the previous step by 2, and add it to the base length to find the triangle's perimeter. Concluding this example, 5 inches multiplied by 2 equals 10 inches, and 10 inches added to 6 inches equals 16 inches --- the triangle's perimeter is 16 inches.

Tips & Warnings

• For an equilateral triangle, select any side length for the base.

References

• Wolfram MathWorld: Isosceles Triangle
• Math Open Reference: Pythagorean Theorem