Here's how to find the area of a regular hexagon with a perimeter of 72 inches:
1. Find the side length:
* A regular hexagon has 6 equal sides.
* Divide the perimeter by the number of sides to get the length of one side: 72 inches / 6 sides = 12 inches per side.
2. Find the apothem:
* The apothem is the distance from the center of the hexagon to the midpoint of a side.
* In a regular hexagon, the apothem forms a 30-60-90 right triangle with half the side length as the base.
* Use the ratio in a 30-60-90 triangle: the apothem is equal to the short leg (half the side length) times the square root of 3.
* Apothem = (12 inches / 2) * √3 = 6√3 inches
3. Calculate the area:
* Area of a regular hexagon = (apothem * perimeter) / 2
* Area = (6√3 inches * 72 inches) / 2
* Area ≈ 374 square inches (rounded to the nearest square inch)
Answer: The area of the hexagon is approximately 374 square inches.
