To find the area of the pentagon picture frame, we need to know the apothem length, which is the distance from the center of the pentagon to the midpoint of one of its sides. For a regular pentagon (all sides and angles are equal), there is a formula to calculate the apothem length:
Apothem length = Side length / (2 * tan(π/5))
Given that the side length of the pentagon is 9 inches, we can substitute this value into the formula to find the apothem length.
Apothem length = 9 / (2 * tan(π/5))
Once we have the apothem length, we can then use the formula for the area of a regular pentagon:
Area = (Perimeter * Apothem) / 2
In this case, the perimeter of the pentagon is simply the sum of all five side lengths (5 * 9 inches). After calculating the area using the formula, remember to round your answer to the nearest tenth as requested in the question.
By following these steps, you can accurately determine the area that Darren's picture frame, shaped like a regular pentagon with 9-inch side lengths, will take up on his wall.
