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.
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.
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.
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.
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:
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Based on this, we can use the Pythagorean theorem, which states that if a and b are the legs of a right triangle and c is the hypotenuse (or the longest side), then a2 + b2 = c2. We can use this to find the height.
Now take a look at either of the right triangles in our sticker. We see that it has legs of lengths 3cm. and h cm., and hypotenuse with length 8cm. Therefore, we can plug this into the Pythagorean theorem to get the following equation:
32 + h2 = 82
Great, now we can solve for h in this equation, and we have our height!
h equals the square root of 55, but that's a really long number and keeps repeating. In this case, we round the square root of 55, so we will say that the height is approximately equal to 7.4, which is what the squiggly equal sign means.
Through this process, we get that h ≈ 7.4 centimeters. See, that's not so bad! In general, to find the height of an isosceles triangle, we simply divide the triangle into two right triangles with the line representing the height, and then we use the Pythagorean theorem to find the height. Easy peasy, and we can now find the area of your sticker! We simply plug b = 6 and h = 7.4 into the area formula, like this:
A = (1/2) ⋅ 6 ⋅ 7.4
A = 22.2
We get that the area of your sticker is 22.2 square centimeters, and now you know how much space it will take up on your windshield. Great!
When it comes to finding the area of an isosceles triangle, it's as simple as plugging values into the area formula and solving. Good thing we are now familiar with this formula. What's better is that we also know how to find the height of an isosceles triangle using the Pythagorean theorem if we don't have it to find the area. Simply divide the base in half, and use it as one of the values in the theorem. Plug the resulting value into the formula as usual, and we're all set!
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