Another Radions/Arcs/Circles problem

juliak · #1 $Old$ November 9th, 2009, 03:58 AM

A chord of a circle subtends an angle of theta radians at the centre of the circle. The area of the minor segment cut off by the chord is 1/8 of the area of the circle.

Prove that 4theta= pi + 4 sin theta

Ahhh i have no idea how to do it?

Grandad · #2 $Old$ November 9th, 2009, 05:57 AM

Hello juliak

Quote:

Originally Posted by juliak $View Post$

A chord of a circle subtends an angle of theta radians at the centre of the circle. The area of the minor segment cut off by the chord is 1/8 of the area of the circle.

Prove that 4theta= pi + 4 sin theta

Ahhh i have no idea how to do it?

If the radius of the circle is $r$ , the area of the sector that makes an angle $\theta$ at the centre is $\tfrac12r^2\theta$ . The area of the triangle with two sides of length $r$ enclosing an angle of $\theta$ is $\tfrac12r^2\sin\theta$ .

So the area of the segment $= ... - ... = \tfrac18\pi r^2$

Fill in the gaps; multiply both sides by $8$ , divide by $r$ and you're there.

Grandad