Area of a segment in a circle

loizoud94 · #1 $Old$ November 30th, 2008, 12:51 PM

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?

Once again, thanks in advance,
D Loizou

loizoud94 · #2 $Old$ November 30th, 2008, 01:18 PM

Nobody?

Soroban · #3 $Old$ November 30th, 2008, 01:24 PM

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: . $\pi(4^2) \:=\:16\pi$ 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: . $\tfrac{1}{3}\pi r^2 \:=\:\tfrac{16}{3}\pi$

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

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

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

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