Using Sine to Find the Area of a Triangle

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  • 0:01 Area of a Triangle
  • 1:28 Using Sine to Find the Area
  • 2:45 Example One
  • 3:53 Example Two
  • 4:47 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Since you first started working with triangles in math class, chances are you've been exposed to the formula for the area of a triangle. However, that is only good if you know the height of the triangle. This lesson shows you how to get around that.

Area of a Triangle

Real fast, how do you find the area of a triangle? If you've been taking geometry for long, you know it's A = 1/2b * h. For many people, this equation comes to mind so quickly that you don't really have to think about it. In fact, it's often something that makes a lot of sense - draw a diagonal line down a rectangle, and you get two equally sized triangles, so you simply find the area of the rectangle and then divide by two. Easy, right?

Well, not always. Chances are those triangles that you've calculated with one half base times height almost always include either a right angle or the height marked. In those instances, it's pretty straightforward to calculate the area. But what about those pesky triangles with no such help? Are we consigned to never knowing their area? Of course not! However, it is going to take a little bit of trigonometry. Namely, we're going to have to use a different formula. In these cases, the area of a triangle is = ½ * a * b * sinC.

The lower case letters are the lengths of the sides and the upper case letter is the measure of the angle indicated. More specifically, C is the angle in between sides a and b. While it may not look as pretty as one half base times height, in this lesson we'll see it is much more useful.

Using Sine to Find the Area

Before we get concerned about the math of that formula, let's step back and look at it. Really, think about it as two parts. The one half times one side should make sense, as it is lifted straight from our usual formula. But what about that second side times the sine of the third angle? What you are essentially doing there is creating an imaginary right triangle in which we know the hypotenuse and the angle and are trying to solve for the opposite side. As such, to solve for the opposite side we just multiply both sides by the hypotenuse, which gives us the opposite side. After all, this ratio of the opposite side over the hypotenuse is all that a sine is - it changes with each angle because the ratio between sides changes. The opposite side of this imaginary right triangle is the height of our real triangle. As a result, we are still calculating one half base times height, but we're going about it in a slightly different way.

Oh, one quick note about calculators. You can use either a scientific calculator or a graphing calculator to do these problems. However, make sure that you are in degrees mode, signified by DEG often. Otherwise, you will get some really messed up answer.

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