A regular hexagon has a perimeter of 60 m. Find its area. Leave your answer in simplest radical form.

To find the area of any regular shape, its formula is

(1/2) • a • p

Where a is the apothem (a line from the center of a side to the center of the shape) and p is the perimeter.

We have the perimeter: 120

Now to find the apothem. To find it, we’re going to cut any sidelength in half, and the line from that half to the center of the hexagon will form a right triangle. We know the bottom leg (where the shape is) is 10 (since 120/6 = 20, and we cut that in half). All we need is angles. To find the interior angle of any regular shape, we use

(180(n - 2))/n

Plugging in, we get 120 degrees for every side. Since we cut it in half, then the top angle of our right triangle is 60. You know what that means? We’ve just found a 30, 60, 90 triangle.

Using the laws of this special triangle, the length of the apothem is the shortest leg times sqrt(3)

So, 10sqrt(3) is your apothem.

Plug everything in!

(1/2) • a • P

(1/2) • 10sqrt(3) • 120

Your area is 600sqrt(3)

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