PLEASE help! Area of a regular decagon!

Ok so I have to find the area of a regular(congruental sided) decagon and I know that the perimeter is 240 and each side is 24. I looked online and found out that the formula is 1/2(ap), so I guess that would make it 1/2(a*240) But idk how to find a. Please help!

you will form triangles by connecting the center of the decagon and its vertices with line segments. To find the height (apothema a), you have isosceles triangles. Using the fact that the interior angles of the decagon measure 144 degrees, we get that the line segment from a vertex to the decagon center bisects the angle. Thus we get a right triangle of base 12 and base angle of 144/2 = 72 degrees. We get the base being 12 from the fact that the line segment from the center of the decagon to the midpoint of one of the sides of the decagon bisects that side.
So the calculations become tan(72) = x/12 implies x = 12*tan(72) which is approximately 36.93. So your calculation for the area becomes A = 1/2 (240*36.93) which is approximately 4431.86. This is your answer.

Look at a decagon. You can break it into a series of triangles. FOr each of those triangles, you know the length of the base(24, as you state above) AND you know the angles (n-2)*180/n = 8 * 18 = 144. Since the triangle sides to the middle bisect the interior angles, these are 72 degrees, and the angle at the centre for each triangle is 36 degrees(if you need it) You can then calculate the height of each trangle (distance from the side to the centre ==> tan(72) = a/12 or a = 12tan(72)

The area of each triangle is 1/2 a * 24 so, multiply this by 10 (yes, 10*24 = 240 = perimeter)

apothem is distance from center to middle of side.

one interior angle of decagon is (10-2)*180/10=144
think of right triangle at bottom of decagon.
a is height
Angle A =144/2=72
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/ | a
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A /__|
12
TanA=a/12
a=12*Tan(72)