What is the area of the regular octagon below?
A regular octagon is shown with a solid segment from the center to the bottom right vertex. The segment is 9 meters. A dashed segment from the center meets the bottom side of the octagon at a right angle.
(1 point)
Responses
114.5 m²
114.5 m²
229.1 m²
229.1 m²
391.1 m²
391.1 m²
458.2 m²
To find the area of a regular octagon, we need to know the length of one of its sides. However, with the given information, we can determine the area of the regular octagon.
The solid segment from the center to the bottom right vertex divides the regular octagon into two congruent trapezoids. The dashed segment from the center to the bottom side divides each trapezoid into a rectangle and a right triangle.
Let's find the length of the side of the octagon. The dashed segment is perpendicular to the bottom side of the octagon, so it bisects the bottom side. Therefore, the length of the bottom side of the octagon is twice the length of the dashed segment.
The dashed segment has a length of 9 meters, so the bottom side of the octagon has a length of 2 * 9 = 18 meters.
Since the octagon is regular, all its sides have the same length. Therefore, each side of the octagon is 18 meters.
Now, let's find the area of each trapezoid. The height of each trapezoid is the dashed segment, which is 9 meters. The bases of each trapezoid are the bottom side of the octagon (18 meters) and the length of one of its sides (18 meters).
The area of each trapezoid is given by the formula:
Area = (1/2) * (base1 + base2) * height
Using the given values, we get:
Area of each trapezoid = (1/2) * (18 + 18) * 9 = (1/2) * 36 * 9 = 18 * 9 = 162 square meters.
Since there are two trapezoids in the octagon, the total area of the octagon is 2 * 162 = 324 square meters.
Therefore, the area of the regular octagon is 324 square meters.
None of the provided answer choices matches the correct answer.