The apothem of a regular polygon is the distance from the center of the polygon to a side. Explain how to derive a formula for the area of a regular polygon using the apothem.
A = N((L/2)*a)..
A = Area of polygon.
N = The number of sides.
L = The Length of a side.
a = apothem or height of a triangle.
To derive a formula for the area of a regular polygon using the apothem, we can break down the process into simpler steps.
Step 1: Start with a regular polygon with n sides, where the length of each side is s.
Step 2: Draw a line segment from the center of the polygon to a vertex. This line segment is called the apothem (a).
Step 3: Now, we need to find the length of the apothem. To do this, we can use trigonometry. Divide the polygon into n congruent triangles, each with a central angle of 360 degrees / n. The apothem is the adjacent side to this central angle.
Step 4: Use trigonometry to find the length of the apothem. We can use the formula for the cosine of an angle: cos(central angle) = adjacent side / hypotenuse. In this case, the adjacent side is the apothem and the hypotenuse is half the length of a side (s/2), as it goes from the center of the polygon to the midpoint of a side.
So, we have: cos(360 degrees / n) = a / (s/2)
Simplifying, we get: a = (s/2) * cos(360 degrees / n)
Step 5: Once we have the length of the apothem, we can calculate the area of each triangle formed by the apothem and two sides using the formula for the area of a triangle: Area = (base * height) / 2. In this case, the base is s, the length of a side, and the height is a, the length of the apothem.
So, the area of each triangle is: Area = (s * a) / 2
Step 6: Since there are n congruent triangles in the regular polygon, we multiply the area of one triangle by n to find the total area of the polygon.
Therefore, the area of a regular polygon with n sides and apothem a is: Area = (n * s * a) / 2
In summary, the formula for the area of a regular polygon using the apothem is (n * s * a) / 2, where n is the number of sides, s is the length of each side, and a is the length of the apothem.