The area of a regular polygon is half the product of the apothem and the perimeter. Let's first identify the apothem and then the side length to obtain the perimeter. Finally, we will use this information to find the area.
Let's do it!
Apothem
An apothem is a segment from the center of the regular polygon perpendicular to a side. We have been given that the length of the apothem of the pentagon is 6.5.
Perimeter
Let's consider the right triangle formed by the radius, the apothem, and the side of the pentagon.
We can find the length of the missing leg by using the Pythagorean Theorem.
a^2+b^2=c^2
Let's substitute the given values into the formula and solve the resulting equation.
Since a is the side of a triangle it must be non-negative, which is why we only kept the principal root when solving the equation. The apothem bisects the side of the regular hexagon. 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 2sqrt(21.75). Since this polygon has five congruent sides, to find its perimeter we will multiply the side length by 5.
Perimeter: 5* 2sqrt(21.75)=10sqrt(21.75)
Area
Now we know that the apothem of the figure is 6.5 and that the perimeter is 10sqrt(21.75). To find its area, we will substitute these values in the formula A= 12aP. Let's do it!
