What is the area of a regular octagon inscribed in a circle of radius $r=6$ meters?

• A. $116.2$ square meters
• B. $288.0$ square meters
• C. $226.2$ square meters
• D. $173.8$ square meters

Answer: (D)

Verified step by step guidance

1. Recall that a regular octagon has 8 equal sides and can be divided into 8 identical isosceles triangles, each with its vertex at the center of the circumscribed circle.
2. Calculate the central angle of each triangle by dividing 360° by 8, which gives the angle \( \theta = \frac{360^\circ}{8} = 45^\circ \).
3. Use the radius \( r = 6 \) meters as the two equal sides of each isosceles triangle. The area of one triangle can be found using the formula for the area of a triangle given two sides and the included angle: \[ \text{Area} = \frac{1}{2} r^2 \sin(\theta) \].
4. Calculate the area of one triangle by substituting \( r = 6 \) and \( \theta = 45^\circ \) into the formula: \[ \text{Area}_{triangle} = \frac{1}{2} \times 6^2 \times \sin(45^\circ) \].
5. Multiply the area of one triangle by 8 to get the total area of the regular octagon: \[ \text{Area}_{octagon} = 8 \times \text{Area}_{triangle} \].