Circle Sector Area

Date: 03/20/2003 at 21:40:05
From: Trevor
Subject: Mandatory conversion

Is it mandatory to convert degrees to radians when finding the area 
of a sector of a circle?

Example:
Find the area of a sector of a circle of radius 4 that is intercepted 
by a 150-degree central angle.

It doesn't say it has to be in radians.

A = (1/2) r^2 theta

.5*16*150 = 1200 units

or

.5*16*(150 Pi/180)= 20Pi/3

Date: 03/20/2003 at 23:02:51
From: Doctor Peterson
Subject: Re: Mandatory conversion

Hi, Trevor.

There is a formula for sector area in terms of degree measure, but it 
is essentially a conversion to radians - in fact, that's what your 
last line looks like. If you use the formula that takes radians, then 
you have to use radians.

Here is the degree-based formula. Since the area of a sector is 
proportional to its central angle, and the area of a whole circle is 
pi r^2, the area of a sector with central angle T degrees (out of 360 
for the whole circle) is

    A = pi r^2 * T/360 = pi/360 r^2 T

If T is in radians, then the formula is

    A = pi r^2 * T/(2 pi) = 1/2 r^2 T

The same is true of the arc length formula. In degrees,

    s = 2 pi r * T/360 = pi/180 r T

and in radians, it is

    s = 2 pi r * T/(2 pi) = r T

The latter is the main reason for using radians at all.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/