# Circle problem - perimater of a minor sector

1. Feb 3, 2005

### footprints

Code (Text):
..
A .'    '.B
. \    / .
.   \  /   .
.    \/    .
.   o    .
.      .
' . .'

The radius of the circle is 5. The perimeter of the minor sector AOB is $$P{\pi} + Q$$. Find P and Q

Last edited: Feb 4, 2005
2. Feb 3, 2005

### HallsofIvy

Staff Emeritus
The circumference of the circle is, of course, $10\pi$. I don't see any way of determining P and Q without knowing what part of the entire circle AOB is. Are you given the central angle? Surely there must be some conditions on P and Q? If P and Q could be any real numbers, then even if we know exactly what the perimeter is, we could choose one of P and Q to be anything we want and then solve for the other.

I take it by "perimeter" you mean the whole perimeter, both the curved part and the two radii. A plausible answer would be that Q= 10 (the length of the two radii) and P would be $\frac{5\theta}{\pi}$.

3. Feb 3, 2005

### saltydog

$$P \pi +Q=5\theta$$

with theta being the radian measure of the angle (inside the slice of pie):

Thus for "any" Q:

$$P=\frac {5\theta-Q}{\pi}$$

Is is really for any Q?

Last edited: Feb 3, 2005
4. Feb 3, 2005

### HallsofIvy

Staff Emeritus
Why "$5\theta$"?

5. Feb 3, 2005

### saltydog

Ohhh, they mean the whole perimeter around the slice of pie and not just the arc length. In that case may I suggest:

$$P \pi +Q=(5\theta+10)$$
$$P=\frac {5\theta+10-Q}{\pi}$$