Answer:
\(54\sqrt{3}\)
Step-by-step explanation:
In order to solve a regular polygon, you would use the equation: \(A=\frac{1}{2} (a)(p)\)
First start by replacing the variables in that equation with what we know. We know that the radius is 6 inches and that hexagons are made up of equilateral triangles. If a 6 inch radius is the side of a single equilateral triangle, then all the sides on that triangle = 6. If all the sides on that one triangle = 6, then all the sides on all the triangles = 6. Therefore, each side of the hexagon = 6.
Now we know that the perimeter = 36. Now our equation is \(A=\frac{1}{2} (a)(36)\).
The apothem is \(3\sqrt{3}\) because the apothem, side, and radius create a right triangle. We know that the radius and side = 6, and if the apothem divides the side in half then the short leg of our triangle = 3. If the length of hypotenuse of our right triangle = 6 and the length of the short leg = 3, then we can use Pythagorean theorem to find the long leg aka. the apothem.
Our equation would then be \(3^{2} +b^{2} =6^{2}\) and so \(b=3\sqrt{3}\).
Now we know what our apothem is so our equation is now \(A=\frac{1}{2} (3\sqrt{3} )(36)\)
Therefore \(A=54\sqrt{3}\)
