Question

find the area of the regular polygon. round your answer to the nearest tenth.
image of a regular octagon with a radius (distance from center to vertex) of 13.07 m and a side length of 10 m
options:

• 966.1 in²
• 483.0 in²
• 80.0 in³
• 176.6 in²

Explanation:

Step1: Identify polygon type

It is a regular octagon (8 sides).

Step2: Recall area formula

Area = $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$

Step3: Calculate perimeter

Perimeter = $8 \times 10 = 80$ m

Step4: Plug values into formula

Area = $\frac{1}{2} \times 80 \times 13.07 = 40 \times 13.07 = 522.8$? Wait, no—wait, the apothem is the distance from center to a side. Wait, the given 13.07 m: if the side length is 10 m, for regular octagon, apothem $a = \frac{s}{2 \tan(\pi/8)} \approx \frac{10}{2 \times 0.4142} \approx 12.07$ m, close to 13.07? Wait, no, maybe the 13.07 is the radius (distance from center to vertex). Let's use radius: side length $s = 2r \sin(\pi/n) = 2 \times 13.07 \times \sin(\pi/8) \approx 26.14 \times 0.3827 \approx 10$ m (matches). Then area of regular polygon: $A = \frac{1}{2} n r^2 \sin(2\pi/n)$

Step5: Calculate with radius

$A = \frac{1}{2} \times 8 \times (13.07)^2 \times \sin(45^\circ)$
$\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071$
$(13.07)^2 \approx 170.82$
$A = 4 \times 170.82 \times 0.7071 \approx 4 \times 120.8 \approx 483.2$

Step6: Round to nearest[SSE Completed, Client Connection Error][SSE onError error]

Answer:

483.0 m² (Note: The options have unit "in²" which may be a typo; the calculation uses given metric units)