How to Find the Area of an Isosceles Triangle

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  • 0:03 The Basics
  • 1:04 The Formula
  • 1:58 Finding the Height
  • 5:26 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions, such as Grand Rapids Community College, Pikes Peak Community College, and Austin Peay State University.

In this lesson, we'll learn how to find the area of an isosceles triangle. We'll also look at how to find the height of an isosceles triangle given the lengths of its sides, since this is needed to find the area.

The Basics

Because we are learning how to find the area of an isosceles triangle, it would be useful to define an isosceles triangle. An isosceles triangle is a triangle with two sides of equal length, like the one shown here.


areaistri1


In general, when it comes to a triangle, we have a nice formula that we can use to find its area. Thankfully, this formula holds true for all triangles. However, before we get to the formula, we should first familiarize ourselves with the different parts of an isosceles triangle.

In an isosceles triangle, we call the two equal sides the legs of the triangle, and we call the other side the base of the triangle. The points at which the sides of the triangle meet are called vertices, and the length from the center of the base to the vertex (single of vertices) opposite the base is called the height of the triangle.

The Formula

Alright, now that we've got the parts down, let's concentrate on the whole. To find the area of an isosceles triangle, we use the following formula:

where A is the area:

A = (1/2)bh

  • b is the base
  • h is the height of the triangle

For example, if we had an isosceles triangle with a base of 8 centimeters and a height of 10 centimeters, we would plug b = 8 and h = 10 into the formula to find the area, like this:

A = (1/2)bh

A = (1/2) ⋅ 8 ⋅ 10

A = 40

We see that the area of an isosceles triangle with base 8 centimeters and height 10 centimeters is 40 square centimeters.

Finding the Height

We now know that finding the area of an isosceles triangle is the same as finding the area of any triangle - we use the formula for the area of a triangle, A = (1/2)bh. However, we often aren't given the height of an isosceles triangle, and we definitely need this in order to use the formula. It is much more common that we would have the lengths of the legs and base of an isosceles triangle than it is that we would have the height.

For example, suppose you buy a state park sticker for the windshield of your car that is in the shape of an isosceles triangle. The sticker gives the lengths of its sides as 8cm, 8cm and 6cm. It doesn't give the height or the area of the sticker, and you want to find the area to know how much space it's going to take up on your windshield. You have the length of the base as 6 centimeters, but in order to find the area, you must find the height of the sticker, so let's look at how to do this.

In general, the line representing the height of an isosceles triangle intersects the base in such a way that it cuts the base in half. This splits the triangle into two right triangles, like in our sticker here:


areaistri5


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