1. ## A Circle Circumscribed about a hexagon

i need your help again !
God bless you more .

*A circle is circumscribed about a hexagon. The area outside the hexagon but inside the circle is 15 square meters.

a) compute the radius of the circle.
b)compute the area of the hexagon.

2. Originally Posted by arenkun
i need your help again !
God bless you more .

*A circle is circumscribed about a hexagon. The area outside the hexagon but inside the circle is 15 square meters.

a) compute the radius of the circle.
b)compute the area of the hexagon.
HI

I assume its a regular hexagon .

So the angle each angle at the centre would be 60 degree .

The formula for area of a segment , A=\frac{1}{2}r^2(\theta-\sin \theta)

Since there are 6 segments , $\displaystyle 15=6(\frac{1}{2}\times r^2(\frac{\pi}{3}-\frac{\sqrt{3}}{2}))$
evaluate r which is approximately 5 m .

Then the area of the hexagon would be the sum of all the 6 triangles ,

Area$\displaystyle =6(\frac{1}{2}\times 5\times 5\times \sin 60)$

3. Hello, arenkun!

A circle is circumscribed about a hexagon.

The area outside the hexagon but inside the circle is 15 mē.

(a) Compute the radius of the circle.

(b) Compute the area of the hexagon.
Code:
                A
* o *
*           *
*               *
F o                 o B

*                   *
*         *         *
*         O         *

E o                 o C
*               *
*           *
* o *
D

The center of the circle is $\displaystyle O.$
The vertices of the hexagon are: .$\displaystyle A,B,C,D,E,F.$
Draw chords: .$\displaystyle AB, BC, CD, DE, EF, FA$
. . .and radii: .$\displaystyle OA, OB, OC, OD, OE, OF$

Assuming it is a regular hexagon, all the chords and radii are equal to the radius $\displaystyle r.$

The area of the circle is: .$\displaystyle \pi r^2$

The hexagon is comprised of six congruent equilateral triangles of side $\displaystyle r.$
The area of one triangle is: .$\displaystyle \frac{\sqrt{3}}{4}r^2$
The area of the hexagon is: .$\displaystyle 6 \times\frac{\sqrt{3}}{4}r^2 \:=\:\frac{3\sqrt{3}}{2}\,r^2$

The difference of the areas is 15 mē: . $\displaystyle \pi r^2 - \frac{3\sqrt{3}}{2}r^2 \:=\:15$

. . $\displaystyle \left(\pi - \frac{3\sqrt{3}}{2}\right)r^2 \:=\:15 \quad\Rightarrow\quad \left(\frac{2\pi - 3\sqrt{3}}{2}\right)r^2 \:=\:15$

Hence: .$\displaystyle r^2 \:=\:\frac{30}{2\pi-3\sqrt{3}} \quad\Rightarrow\quad r \;=\;\sqrt{\frac{30}{2\pi-3\sqrt{3}}}$

Therefore: .$\displaystyle r \;=\;5.253385683\text{ m} \;\;(a)$

The area of the hexagon is: . $\displaystyle \frac{3\sqrt{3}}{2}\,r^2 \;=\;\frac{3\sqrt{3}}{2}\left(\frac{30}{2\pi-3\sqrt{3}}\right) \;=\;71.70186611\text{ m}^2\;\;(b)$

4. TO : Math Addict and Soroban

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### Formula on how to determine the area of the hexagon if the area outside the hexagon but inside the circle is 15 sq.cm?

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