Question

Find the are of the regular hexagon PLZ help

Answer 1

Check the picture below.

let's recall that a circle has a total of 360°, and if we draw two radii from the center of the hexagon, the angle at the center made by those two radii will be 60°, because the hexagon will split the 360° in 6 even pieces, 360/6 = 60.

anyhow, if we use half of that triangle made by two radii, we'll end up with a 30-60-90 triangle, as you see in the picture, and thus we can use the 30-60-90 rule.

so, when doing so, notice, each side of the hexagon is 8 units long, therefore, the perimeter of the hexagon is then 8*6 = 48.

`\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{2}ap~~ \begin{cases} a=apothem\\ p=perimeter\\[-0.5em] \hrulefill\\ a=4√(3)\\ p=48 \end{cases}\implies A=\cfrac{1}{2}(4√(3))(48)\implies A=96√(3) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill A\approx 166.28~\hfill`

Answer 2

166.3

Explanation:

A hexagon consists of six equilateral triangles, and we can divide each of them into two right triangles.

So, we can calculate the area of one right triangle and multiply by 12.

The formula for the area of one triangle is

A₁ = ½bh

We can use the Pythagorean Theorem to calculate the base of one of the small triangles.

b² + (4√3)² = 8²

b² + 16 × 3 = 64

b² + 48 = 64

b² = 16

b = 4

The area of one small triangle is

A = ½ ×4 × 4√3 = 8√3

So, the area of 12 small triangles is

A = 12 × 8√3 = 96√3 ≈ 166.3

The area of the hexagon is 166.3.