Question

the area of the octagon below is 10,391.9 square millimeters.
what is the approximate perimeter of the octagon?
round to the nearest whole number.
$\boldsymbol{square}$ mm
what is the approximate length of one side of the octagon? round to the nearest whole number.
$\boldsymbol{square}$ mm
56 mm

Explanation:

Step1: Recall area formula for regular polygon

The area of a regular polygon is $A = \frac{1}{2} \times a \times P$, where $a$ is the apothem (56 mm), $A$ is the area (10,391.9 mm²), and $P$ is the perimeter.

Step2: Rearrange to solve for perimeter

Rearrange the formula to isolate $P$: $P = \frac{2A}{a}$
Substitute values: $P = \frac{2 \times 10391.9}{56}$
Calculate: $P = \frac{20783.8}{56} \approx 371.139$
Round to nearest whole number: $P \approx 371$

Step3: Find length of one side

A regular octagon has 8 equal sides. Side length $s = \frac{P}{8}$
Substitute $P \approx 371$: $s = \frac{371}{8} = 46.375$
Round to nearest whole number: $s \approx 46$

Answer:

Approximate perimeter: 371 mm
Approximate length of one side: 46 mm