Question Video: Deriving a Formula for Calculating the Area of a Sector When Using Radians

Write an expression for the area of a sector whose arc’s measure is πœƒ radians, knowing that the expression for the area of a sector measuring πœƒ degrees is πœ‹π‘ŸΒ²πœƒ/360.

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Video Transcript

Write an expression for the area of a sector whose arc’s measure is πœƒ radians, knowing that the expression for the area of a sector measuring πœƒ degrees is πœ‹π‘Ÿ squared πœƒ over 360.

So we’re reminded of the formula we can use to calculate the area of a sector when the central angle is given in degrees. And we’re asked to use this to determine a different formula we can use when the angle is given in radians. We should recall that when we’re working in radians, a full turn, which in degrees is equivalent to 360 degrees, is two πœ‹ radians. So we can take the formula that we know for the area of a sector in degrees, and we can replace the 360 in the denominator, which represents the 360 degrees in a full turn with two πœ‹. Doing so gives πœ‹π‘Ÿ squared πœƒ over two πœ‹. Now, of course, we can cancel a factor of πœ‹ from the numerator and denominator of this fraction, which leaves us with π‘Ÿ squared πœƒ over two or, equivalently, one-half π‘Ÿ squared πœƒ.

So we’ve used the area of a sector in degrees to find an expression for the area of sector when the central angle is given in radians; it’s one-half π‘Ÿ square πœƒ.

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