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How to Use Trigonometry to Find the Area of a Triangle

Sometimes, it’s hard to find the area of a triangle, especially when it’s not convenient to measure its height. When this happens, you can use trig to modify this traditional formula for the area of a triangle,

image0.png

and help you calculate the area. In this formula, A is the area, b is the length of the triangle’s base, and h is the height of the triangle drawn perpendicular to that base. The following figure shows a triangle where the traditional formula applies.

image1.jpg

This formula works fine if you can be sure that you’ve measured a height that’s perpendicular to the side of the triangle. But what if you have a triangular yard — a big triangular yard — and have no way of measuring some perpendicular segment to one of the sides?

Well, you can use trigonometry — or, at least, a formula with an angle measure in it. To measure that angle, you can be very sophisticated and get a surveying apparatus, or if you’ve got a protractor handy, you can do a decent estimate by extending the sides at an angle for a bit and eyeballing the angle size.

So, using trig, you can modify the traditional formula for finding the area of a triangle, as follows:

image2.png

where a and b are two sides of the triangle and is the angle formed between those two sides. You don’t need the measure of the third side at all, and you certainly don’t need a perpendicular side.

image3.jpg

Here’s where this formula comes from: first, take a look at the triangle in the preceding figure, with sides a and b and the angle between them.

Start with the traditional formula for the area of this triangle,

image4.png

Then look at the little triangle to the left. Because the height is perpendicular to the base, the sides and height form a right triangle. The acute angle θ has a sine equivalent to the following:

image5.png

If you solve that equation for h by multiplying each side by a, you get

image6.png

Replace the h in the traditional formula with its equivalent from the preceding formula, and you get

image7.png

Now you’ll see how this formula works in an actual problem. The triangle in the following figure shows the measures of two of its sides and the angle between them.

image8.jpg

To find the area of this triangle, follow these steps:

  1. Use the trig formula, inserting the values that you know.

    image9.png
  2. Solve for the value of the area.

    image10.png

    The area is about 8660 square units.

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