Question

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Use the formula for area of a triangle to derive the formula for the area of a regular polygon.

Answer 1

Draw triangles by using the outside edges as the bases of the triangles and lines going to the middle of the regular polygon as the other sides of the triangles. Use Pythagorean formula to get height of these triangles and then areas = (1/2)(base)(height).

Answer 2

Think of an n sided polygon as n triangles.
The area of each triangle is the length of one side * the distance from the midpoint to the centre of the polygon /2.
So, the area of the polygon is n * length of one side * the distance from the midpoint to the centre of the polygon /2.
n * length of one side = perimeter
distance from the midpoint to the centre of the polygon = apothem

So area = apothem*perimeter/2