the apothem of a regular polygon is the distance from the center of the polygon to a side. Explaiin how to derive a formula for the area of a regular polygon using the apothem.
Well, that apothem seems to be the altitude of one of n triangles.
The area of each of those triangles is
(1/2)* apothem * length of a side
so the total area would be
(n/2)*apothem * length of a side
To derive a formula for the area of a regular polygon using the apothem, we need to break down the process step by step. Here's how you can do it:
1. Start with a regular polygon: A regular polygon is a polygon with all sides and angles congruent (equal) to each other.
2. Draw the apothem: The apothem is a line segment drawn from the center of the polygon to a side, perpendicular to that side. Label the apothem length as 'a'.
3. Find the length of each side: To derive the formula, it's essential to determine the length of each side of the regular polygon. Let's label it as 's'.
4. Divide the polygon into congruent triangles: Draw lines connecting the center of the polygon to each vertex. This will divide the regular polygon into congruent triangles.
5. Find the height of each triangle: The height of each triangle is the length of the apothem ('a'). As the sides of the regular polygon are congruent, the height will be perpendicular to the base (a side).
6. Find the base of each triangle: The base of each triangle is equal to the length of a side ('s').
7. Calculate the area of each triangle: The area of a triangle can be calculated using the formula: A = (base * height) / 2. In this case, the base is 's' and the height is 'a'. Hence, the area of each triangle is (s * a) / 2.
8. Determine the total number of triangles: The total number of congruent triangles in a regular polygon is equal to the number of sides, denoted by 'n'.
9. Calculate the total area of the regular polygon: To find the area of the regular polygon, multiply the area of each triangle calculated in step 7 by the number of triangles (n). So, the area of the polygon will be: Total Area = (n * (s * a)) / 2.
By following these steps, you can derive a formula for the area of a regular polygon using the length of the apothem.