If you divide the pentagon into congruent triangles, you can quickly find the area of the shape. Draw a radius from the center of the circle to each corner of the pentagon. These radii divide the pentagon into five isosceles triangles each with a center angle of 360/5 = 72 degrees (once around the circle, divided by five triangles) and two sides of length 8 cm. You can find the length of the third side in one of two ways.
- Using the law of cosines.
- Draw a perpendicular from the center of the circle to the third side of the triangle and use the sine and cosine of 72/2 = 36 degrees.
Can you see the next step? you have five copies of an isosceles triangle and you know all the side lengths, so you should be able to find the area of the triangle and therefore, the whole pentagon.
Hope this helps,
Stephen and Penny