Question 31876: How do I find the area, radius and apothem of a regular hexagon with the side length of 18?
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! The apothem (r) of a regular polygon is given by: The apothem is the perpendicular line from the centre of the regular polygon to one of the sides. This is also the radius of the inscribed circle.
Where: r = apothem, s = length of the side, n = number of sides.
The area (K) of a regular polygon is given by:
The radius (R) of a regular polygon is given by: Here, we understand the radius (R) of the regular polygon to be the radius of the circumscribed circle.
The apothem (r) is:
The area (K) is:
The radius (R) is:
There is an easier method that doesn't require the use of trigonometry which I should have thought of first, but it was late and past my bed time.
If you were to draw the three diagonals of the regular hexgon, you will have created six equilateral triangles inside the figure. If you will accept this much, and it's easy enough to prove, then we can proceed.
Let's look at one of the six equilateral triangles whose sides measure 18 each.
The radius of the hexagon is just one of the sides so it's 18.
The apothem is the height of one of the triangles which can be found using the Pythagorean theorem  .
If you draw the apothem from the center of the hexagon, perpendicular to one side of the hexagon, you'll see that it bisects that side thus creating two right triangles from one of the equilateral triangles.
Let's call the apothem r, as we did in the first solution, then the right triangle has sides r, R = 18, and a side of 18/2 = 9. Applying Pythagorus:
 Solve for r.
 Take the square root of both sides.
Now, for the area (K) of the hexagon, we'll find the area of one of the six equilateral triangles, then multiply by 6 to get the entire area.
 Now multiply by 6.
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