Question

1. Use the diagram of the regular pentagon to support an explanation showing why the formula accurately yields the area of the pentagon. (Recall that a is the apothem and P is the perimeter of the pentagon.)

Answer 1

(Please check the attachment for the diagram which will support the answer)

The regular pentagon is made up of five triangles as shown in the diagram. Let us now take the case of the bottom triangle ODC. We know from basic geometry that the area of any triangle is half times base times height. So, the area of the triangle will be half times DC (which is the base) times OP (which is the height).

Now, if we take into account all the triangles that make the pentagon we will get similar areas. The height of all such triangles will remain the same as OP because this is a regular pentagon.

Adding all these areas will give us the area of the pentagon which will come out to be:

Area of Pentagon=Area of Triangle OCD + Area of Triangle OBC +Area of Triangle OAB + Area of Triangle OAE + Area of Triangle OED

(Please see attachments for complete solution)

As we can see from the last attachment, the sum in the parenthesis is the perimeter of the pentagon and thus we get the area of the pentagon as

1/2 times apothem times Perimeter