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
mm
what is the approximate length of one side of the
octagon? round to the nearest whole number
mm
23
46
371
1,299

Explanation:

Step1: Recall the formula for the area of a regular polygon

The formula for the area of a regular polygon is \(A=\frac{1}{2}aP\), where \(A\) is the area, \(a\) is the apothem, and \(P\) is the perimeter.

Step2: Solve for the perimeter \(P\)

We are given that \(A = 10391.9\) square - millimeters and \(a=56\) mm.
From \(A=\frac{1}{2}aP\), we can solve for \(P\) by re - arranging the formula: \(P=\frac{2A}{a}\).
Substitute \(A = 10391.9\) and \(a = 56\) into the formula:
\(P=\frac{2\times10391.9}{56}\)
\(P=\frac{20783.8}{56}\)
\(P\approx371\) mm.

Step3: Solve for the side length \(s\)

For a regular octagon (a polygon with \(n = 8\) sides), the perimeter \(P=ns\), where \(s\) is the side length.
We know \(P\approx371\) and \(n = 8\).
From \(P=ns\), we solve for \(s\) by \(s=\frac{P}{n}\).
Substitute \(P = 371\) and \(n = 8\) into the formula: \(s=\frac{371}{8}=46.375\approx46\) mm.

Answer:

The approximate perimeter of the octagon is \(371\) mm.
The approximate length of one side of the octagon is \(46\) mm.