An apothem is a segment from the center of the regular polygon perpendicular to a side. In the diagram we see that the length of the apothem is 2.5
Perimeter
Consider the right triangle formed by the apothem, the radius and the side of the heptagon.
We find the length of the side b using the Pythagorean Theorem.
a^2+b^2=c^2
Let's substitute the corresponding values into the formula and solve the resulting equation.
Since b is the side of a triangle it must be non-negative, which is why we only kept the principal root when solving the equation. Recall that the apothem bisects the side of the regular heptagon. Therefore, the length of the side of the given polygon is twice the length of the side of the above triangle.
Consequently, the side length of the regular hexagon is 2b=2sqrt(1.4229). Since this polygon has seven congruent sides, to find its perimeter we will multiply the side length by 7.
Perimeter: 7* 2sqrt(1.4229)=14sqrt(1.4229)
Area
Now that we know that the apothem of the figure is 2.5 and that the perimeter is 14sqrt(1.4229). To find its area, we will substitute these values in the formula A= 12aP. Let's do it!
