Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
If radii are drawn from the center of a regular polygon to the vertices, congruent isosceles triangles are formed. Using the apothem as the height and the polygon side as the base, the area of each triangle can be calculated and summed. Therefore, the area regular polygons is equal to the number of triangles formed by the radii times their height: (side length)(apothem length)(number of sides)/2.
If we want to calculate the area of any regular polygon, the first thing we need to do is to write it inscribed in a circle, and by doing that we've created the ability to draw an apothem. Now an apothem is a perpendicular segment from the center of the circumscribed circle to one of the sides of your polygon. Now the reason why this is important is because if we want to calculate this hexagon, so again this will apply to any regular polygon. What we're going to do is we're going to think, "well Mr. McCall, I have no idea how to calculate that area. But what I do know how to calculate is the area of a triangle." So if I divide this hexagon into congruent triangles, and they'll all be congruent because it's regular, then all I have to do is add up the area of my 6 triangles. So you can say the area of the hexagon, is equal to 6 times the area of one of those triangles. But how do we calculate the area of one of those triangles?
Well, if you recall in order to calculate the area of a triangle, you need two things. You need the base and its corresponding height. So that's why we need the apothem because the apothem is that corresponding height. So specifically for a polygon, we're going to call this a and we're going to call that base x where, if we go back here s is our side length. So notice that the sides will all be the same for a regular polygon. So the area of this triangle is going to be the apothem which is the height. So I'm going to say 6 times the apothem times the base which is s divide by 2. So the area of a hexagon is equal to 6 times the area of one of your triangles. But that's not very useful because this is only going to apply to a hexagon. So let's make this area formula for any type of polygon. So instead of writing 6, I'm going to write n for the number of sides. So I can write this as apothem times side length times number of sides all divided by 2.
Now if we look at this, I see that I can simplify this a little bit more. Getting back to our regular polygon. The perimeter, capital p of this polygon is going to be 1, 2, 3, 4, 5, 6s. If I had a pentagon, that would be 5s. So what I'm going to say that the perimeter of any regular polygon is n times s. So I can substitute in for n times s, capital p which is going to stand for perimeter. So there's going to be two ways to write your area formula and I'm going to erase this to make it a little bit clear here.
So we're going to say the area of any regular polygon is equal to apothem times side length times the number of sides divided by 2, or if we substitute in for n times s, it's going to be the apothem times the perimeter divide by 2. So you have 2 different formulas, both of which will calculate the area of any regular polygon.