Answer:
236.59mm²
Step-by-step explanation:
The formula for the area of a regular polygon is 1/2ap, with <em>a</em> being the apothem and <em>p</em> being the perimeter.
To start, get the perimeter by multiplying 7 (one side of the polygon) by 8 to get 56.
Next, we have to find the apothem (the perpendicular segment from the center of the polygon to one of the sides).
First, create a right triangle on an of the polygon's sides, where the hypotenuse is equal to the radius of the polygon, one of the legs is the apothem, and the other leg is equivalent to half of the side.
Next, using the polygon angle formula (n-2(360)/n), we can get the base angle of the triangle, 1/2(135), or 67.5.
We can use this angle and some trigonometry to find the the apothem.
First, use SOHCAHTOA to figure out if you will use sine, cosine, or tangent (in this case, tangent).
Next, set up an equation using the tangent of 67.5 to find the apothem - tan67.5 = a/3.
And set the equation equal to <em>a</em> (a = 3.5tan67.5).
Now that you have the apothem, plug it into the area formula to get A = 1/2(56)(3.5tan67.5) and solve!
(28)(3.5tan67.5)
(28)(8.4497)
236.59mm²
