What is an Isoceles Triangle? - Geometry Made Easy

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Isoceles Triangles

Isoceles Triangle
Isoceles Triangle

Isoceles triangles have at least two sides that are exactly the same length. This forces two of their angles to also be acute angles of exactly the same size. In this blue triangle, the two longer sides are the same length, which forces the two bottom angles to be the same size. If the third side is also the same length as the first two sides, then the triangle is an equilateral triangle. All equilateral triangles have to be isoceles (eye-SAH-suh-leez) triangles, but not all isoceles triangles are equilateral.

Isoceles triangles, like other triangles, have a perimeter and an area. You can find out the perimeter of an isoceles triangle by adding together the length of all three sides. It's harder to find out the area.

Area of an isoceles triangle

To find the area of an isoceles triangle, start by drawing a line down the middle, from the top point to the middle of the bottom side. You'll notice that an isoceles triangle has bilateral symmetry - that will be useful. Now you have two smaller triangles, but they're exactly the same size, each half the area of the original triangle.

The line you drew down the middle is perpendicular to the bottom side of the triangle, so those two lines meet at right angles, forming two right triangles that are the same size. Because each of these right triangles makes half of a rectangle, you can imagine moving one of them, turning it upside down, and putting it against the other one to make a rectangle. Now it is easy to find the area of that rectangle if you know the height of the isoceles triangle and how wide the bottom is.

But suppose you only know the lengths of the three sides, and not how long the center line is? No problem. You can calculate the height of the center line using the Pythagorean Theorem, because half of an isoceles triangle is a right triangle.

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Bibliography and further reading about geometry:

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Copyright 2012-2015 Karen Carr, Portland State University. This page last updated August 2015.

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