PENTAGONS The area of most figures can be found out by dividing the figure into rectangles and triangles. However, this is not true with irregular pentagons. For regular pentagons, where each side (s) is equal, the area is area = (½) (apothem) (perimeter), where the apothem is the distance from the center to a corner. In an irregular pentagon, it seems that you can divide up into triangles, like with many irregular figures. What I do is I take a corner and connect it to two other corners. Do this with each corner until you get a star-like shape in the middle. You get 10 triangles, which you can find the area of (bh/2). But sadly, you get another pentagon. You can do that with the smaller pentagon, but then you get an even smaller one. This goes on infinitely. As you can see, pentagons give us an idea of infinity. You may notice that the pentagon inside of it is dilated exactly k= ½. For those of us that remember pre-algebra, this means that the figure is exactly half the size of the original pentagon. To calculate the approximate area, you do not need to take an infinite amount of measurements. You actually only need 15. The 5 measures of the sides, (a1, b1, c1, d1, and e1), the 5 heights of the outside triangles, (a2, b2, c2, d2, and e2), and the 5 distances from each corner to the inner pentagon, (f, g, h, i, and j). The lengths of the inner pentagon are equal to half of one of the outer pentagon’s length. A formula for the approximate area of a pentagon is: ∑_(n=1)^∞▒(a_1 a_2+b_1 b_2+c_1 c_2+d_1 d_2+e_1 e_2+a_1/2 f+b_1/2 g+c_1/2 h+d_1/2 i+e_1/2 j)/2n =AREA OF AN IRREGULAR PENTAGON The sigma means that it repeats infinity and n doubles every time Wait for the Approx variable to stop changing to get a good approximation Area of the pentagon shown on the screen is about 202.84 square centimeters
