Question

Find the missing measures in this regular hexagon.

A regular hexagon has side lengths of 16 inches. The radius is 16 inches. An apothem is shown.

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.

Answer 1

Answer:

see explanation

Step-by-step explanation:

The apothem bisects the side at right angles.

let a represent the apothem

Using Pythagoras' identity in the right triangle

The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, thus

a² + 8² = 16², that is

a² + 64 = 256 ( subtract 64 from both sides )

a² = 192 ( take the square root of both sides )

a = $\sqrt{192}$ ≈ 14 in ( to nearest whole number )

The perimeter of the hexagon = 6 × 16 = 96 in

The area (A) of the hexagon is calculated as

A = perimeter × apothem

   = 96 × $\sqrt{192$ ≈ 1330 in² ( to nearest whole number )

Answer 2

The length of the apothem of the hexagon is about 13.856 inches.

The perimeter of the hexagon is 96 inches.

The area of the hexagon is about 665.088 square inches.

Given length of a side is = 16 in

So half side is = 8 in

From the given right triangle using Pythagorean theorem:

Apothem = $\sqrt{16^{2}-8^{2}}=13.856$.

There are 6 sides of hexagon. Each side= 16 in.

So perimeter = 6(16) = 96 in.

The area is:

$\frac{1}{2}\cdot apothem\cdot perimeter\\\frac{1}{2}\cdot13.856\cdot96\\=665.088$

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