# Thread: Finding area of star in circle.

1. ## Finding area of star in circle.

Hi i need to find the area of the star only knowing the radius of the circle: r=1

2. ## Re: Can someone help me to find this area?

Originally Posted by leart369
Hi i need to find the area of the star only knowing the radius of the circle: r=1

Have a look here: Pentagram - Wikipedia, the free encyclopedia

3. ## Re: Can someone help me to find this area?

Originally Posted by earboth
I don't get it

4. ## Re: Can someone help me to find this area?

Hello, leart369!

I have a very "clunky" solution . . .

i need to find the area of the star only knowing the radius of the circle: R=1

Label the vertices $A,B,C,D,E$ starting at the top and moving clockwise.
. . Draw segments $AB,BC,CD,DE,EA.$
Let $O$ be the center.
$\text{Let }P\,=\,AC \cap BE.\;\text{Let }Q \,=\,AD \cap BE.$

Consider the area of one-fifth of the star, quadrilateral $AOBP$

The area of a triangle is given by: . $A \:=\:\tfrac{1}{2}ab\sin C$
. . (one-half the product of two sides and the sine of the included angle)

The area of $\Delta AOB \:=\:\tfrac{1}{2}R^2\sin72^o$

The area of $\Delta APB \:=\:\tfrac{1}{2}a^2\sin108^o$

. . The area of quad $AOBP \:=\:\tfrac{1}{2}R^2\sin72^o - \tfrac{1}{2}a^2\sin108^o$

Therefore: . $\text{Area of star} \:=\:\tfrac{5}{2}\left(R^2\sin72^o - a^2\sin108^o\right)$

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We know that $R = 1.$ . Can we find $a\,?$ . . . . Yes!

$\begin{array}{cccccc}\text{In }\Delta APQ\!: & b \:=\:2a\sin18^o & [1] \\ \text{In }\Delta AOC\!: & 2a+b \:=\:2R\sin72^o & [2] \end{array}$

Substitute [1] into [2]: . $2a + 2a\sin18^o \:=\:2R\sin72^o$

. . $a(1+\sin18^o) \:=\:R\sin72^o \quad\Rightarrow\quad a \:=\:\frac{R\sin72^o}{1+\sin18^o}$

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# area of a.star inside a circle

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