What is the area of a regular hexagon circumscribed iinside a circle with a radius of 1?

Question

What is the area of a regular hexagon circumscribed iinside a circle with a radius of 1?

1 Answer

Impact of this question

14477 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-18T06:05:44

$\frac{3 \sqrt{3}}{2}$ $frac{3sqrt{3}}{2}$

Explanation:

The regular hexagon can be cut into 6 pieces of equilateral triangles with length of 1 unit each.

For each triangle, you can compute the area using either

1) Heron's formula, $Area = \sqrt{s (s - a) (s - b) (s - c)}$ $"Area"=sqrt{s(s-a)(s-b)(s-c)$ , where $s = \frac{3}{2}$ $s=3/2$ is half the perimeter of the triangle, and $a$ $a$ , $b$ $b$ , $c$ $c$ are the length of the sides of the triangles (all 1 in this case). So $Area = \sqrt{(\frac{3}{2}) (\frac{1}{2}) (\frac{1}{2}) (\frac{1}{2})} = \frac{\sqrt{3}}{4}$ $"Area"=sqrt{(3/2)(1/2)(1/2)(1/2)}=sqrt{3}/4$

2) Cutting the triangle in half and applying Pythagoras Theorem to determine the height ( $\frac{\sqrt{3}}{2}$ $sqrt{3}/2$ ), and then use $Area = \frac{1}{2} \cdot Base \cdot Height$ $"Area"=1/2*"Base"*"Height"$

3) $Area = \frac{1}{2} a b sin C = \frac{1}{2} (1) (1) sin (\frac{π}{3}) = \frac{\sqrt{3}}{4}$ $"Area"=1/2 a b sinC=1/2 (1) (1) sin(pi/3)=sqrt{3}/4$ .

The area of the hexagon is 6 times the area of the triangle which is $\frac{3 \sqrt{3}}{2}$ $frac{3sqrt{3}}{2}$ .

Science

Math

Humanities

... and beyond

What is the area of a regular hexagon circumscribed iinside a circle with a radius of 1?

1 Answer

Explanation:

Related questions

Impact of this question