Question

find the missing measures in this regular hexagon. the length of the apothem of the hexagon is about inches. the perimeter of the hexagon is inches. the area of the hexagon is about square inches. 222.4 667.2 1334.4

Explanation:

Step1: Calculate the apothem

For a regular hexagon, if the side length \(s = 16\) inches and the radius \(r=16\) inches.
The apothem \(a\) of a regular hexagon can be found using the formula \(a = r\cos30^{\circ}\) (since the central angle of a regular hexagon is \(60^{\circ}\), and when we split the equilateral - triangle formed by two radii and a side of the hexagon into two right - triangles, the angle between the radius and the apothem is \(30^{\circ}\)).
We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\) and \(r = 16\) inches.
So, \(a=16\times\cos30^{\circ}=16\times\frac{\sqrt{3}}{2}=8\sqrt{3}\approx13.9\) inches.

Step2: Calculate the perimeter

The perimeter \(P\) of a regular hexagon with side length \(s\) is given by \(P = 6s\).
Since \(s = 16\) inches, \(P=6\times16 = 96\) inches.

Step3: Calculate the area

The area \(A\) of a regular polygon is given by \(A=\frac{1}{2}aP\), where \(a\) is the apothem and \(P\) is the perimeter.
We found \(a\approx13.9\) inches and \(P = 96\) inches.
\(A=\frac{1}{2}\times13.9\times96=13.9\times48 = 667.2\) square inches.

Answer:

The length of the apothem of the hexagon is about \(13.9\) inches.
The perimeter of the hexagon is \(96\) inches.
The area of the hexagon is about \(667.2\) square inches.