Area of a segment in a circle

• Nov 30th 2008, 10:51 AM
loizoud94
Area of a segment in a circle
I was wandering if anybody could help with this, it would be fully appreciated:

Two parallel lines are drawn 2cm from the centre of a circle of radius 4cm. Calculate the area between them.

I know to find the area of the circle first[16pi], but where do I go next?

D Loizou
• Nov 30th 2008, 11:18 AM
loizoud94
Nobody?
• Nov 30th 2008, 11:24 AM
Soroban
Hello, loizoud94!

Quote:

Two parallel chords are 2cm from the centre of a circle of radius 4cm.
Calculate the area between them.

Code:

              * * *           *:::::|:::::*         *:::::::|:::::::*       A*- - - - + - - - -*B           *          *       *      * 120°*      *       *        *O        *       *        |        *                 |       C*- - - - + - - - -*D         *:::::::|:::::::*           *:::::|:::::*               * * *

The area of the circle is: .$\displaystyle \pi(4^2) \:=\:16\pi$ cm².

The area between the chords is:
. . the area of the circle minus the area of the two shaded segments.

We find that the sector has a central angle of 120°.
Hence, the area of the sector is: .$\displaystyle \tfrac{1}{3}\pi r^2 \:=\:\tfrac{16}{3}\pi$

The area of $\displaystyle \Delta AOB$ is: .$\displaystyle \tfrac{1}{2}(4)(4)\sin120^o \:=\:4\sqrt{3}$

. . Hence, the area of a segment is: .$\displaystyle \tfrac{16}{3}\pi - 4\sqrt{3}$

The area betwen the chords is: .$\displaystyle A \;=\;16\pi - 2\left(\tfrac{16}{3}\pi - 4\sqrt{3}\right)$

Therefore: .$\displaystyle A \;=\;\tfrac{16}{3}\pi + 8\sqrt{3}$ cm²