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Center and Apothem of Regular Polygons - Concept

Teacher/Instructor Brian McCall
Brian McCall

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

An apothem is a perpendicular segment from the center of a regular polygon to one of the sides. When radii are drawn from the center to the vertices of the polygon, congruent isosceles triangles are formed with the polygon apothem as the height. These triangles are used in calculating the area of regular polygons. Related topics include properties of isosceles triangles and area of triangles.

In order to find the area of any regular polygon, first we need to inscribe it inside a circle. By doing that, we've created this apothem, where the definition of the apothem is a perpendicular segment from the center to the sides of the polygon. But back up. What's the center? Well, the center of this polygon is the center of the circle that circumscribes the polygon. So notice that I can draw in a radius from that center to a vertex and that is also the radius of the circle. So if we zoom in on what's going on with this apothem, the perpendicular segment from the center to one of the sides of the polygon. And if I draw in 2 radii, since the radius of a circle is constant, we've created an isosceles triangle.
So whenever you draw in apothem and segments to the vertices, you are going to create an isosceles triangle. One property of an isosceles triangle is if you have an altitude, you are bisecting that vertex angle. So remember that this apothem will bisect it and you'll have two congruent base angles down here.
Now specifically for a hexagon. Since I can draw in 6 radii, we're going to divide 360 degrees 6 ways, which means each of these angles up here is going to be 60 degrees, and since these two base angles are congruent, we're going to have equilateral triangles. So this only works for a hexagon. But remember when you're problem solving that by drawing in your radii, you're creating 6 equilateral triangles. But this doesn't give us an area formula. So what we're going to have to do is we're going to have to think in terms of this triangle.
Let's say I knew the length of this base, which is also going to be our side length. So I'm going to call that s. I can calculate the area of this triangle by saying, my apothem which is the height of this triangle times the base which is s and then dividing it in half. That's just the definition for the area of a triangle. Is the base times height divided by 2.
The question is how many of these triangles are we going to have? Well, in the case of a hexagon, we're going to have 6 triangles so I would have to multiply this by 6. And what I'm going to do is I'm going to generalise this formula and say that instead of writing 6 just for a hexagon, I'm going to write n for an n-sided polygon. So this formula will only work for a regular polygon. Which means equilateral and equiangular.
So the area formula is going to be the area, the excuse me the apothem times the side length times the number of sides all divided by 2 and what you're doing here is you're calculating the area of one of these triangles and then multiplying it by however many triangles you have.

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