Area of Regular Polygons: Practice Using Trigonometry

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  • 0:01 Any Regular Polygon
  • 0:52 A Note on Circles
  • 1:28 Finding Angles
  • 2:29 Using Trig
  • 3:22 Solving for Area
  • 4:26 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

Tired of memorizing different area formulas that are becoming more and more complex? With a little trigonometry, you'll be able to calculate the area of any regular shape, whether 4 or 40 sides!

Any Regular Polygon

One of the most useful parts about geometry is the ability to solve for the area of a shape, especially a regular shape. However, while the formulas for solving the area of a square or a parallelogram aren't that bad, have you seen the one to solve the area of a hexagon? In case you were curious, it is three times the square root of 3, divided by 2, times the length of a side squared. Now I don't know about you, but that's more than I really care to remember. Wouldn't it be nice if there were a much easier way of doing this?

Luckily, there is. As long as you are taking the area of a regular polygon, which means that all the internal angles are congruent, then you can use trigonometry to solve for the area. Trust me, it's not as scary as it sounds. In fact, in this lesson we're going to use it to solve the area of a regular dodecagon, a 12-sided figure, that has a side length of 8.

A Note on Circles

First of all, remember that all those internal angles are not also equal, but so are the external angles too! In fact, they also all add up to 360 degrees. You can take my word on it, remembering that each angle in a square is 90, and four 90s makes 360. Or you can be a bit more adventurous and draw a circle around the square or the dodecagon. You'll see that your pen turns the same number of degrees around - precisely 360 degrees. That's about to be really important for us when we start finding the interior angles of our dodecagon. Have you drawn it yet? Perfect. Let's move on.

Finding Angles

Alright, here's where things can get a bit complicated. Remember how I said that each of those angles was equal to the others? Because we have 12 sides, that means we have twelve angles of 30 degrees each. However, the angles are not where you may first expect them to be. If you want, go ahead and sketch straight lines from each angle to the center of the dodecagon. Now we end up with twelve triangles, each of which has a 'little' angle of 30 degrees. It is these angles that must add up to 360.

But wait, the interior angles of a triangle don't add up to 360 degrees, they only add up to 180 degrees. You're exactly right! Therefore, each of those triangles has 3 angles, one of which is 30 degrees, and the other two are each 150 divided by two, or 75 degrees. By the way, we got 150 degrees by subtracting 30 from 180. So, now we've got 12 triangles with a short side of 8 and angles of 30, 75, and 75 degrees.

Using Trig

Area of a Dodecagon

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