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

Video Transcript

Write an expression for the area of a sector whose arc’s measure is 𝜃 radians, knowing that the expression for the area of a sector measuring 𝜃 degrees is 𝜋𝑟 squared 𝜃 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 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. Doing so gives 𝜋𝑟 squared 𝜃 over two 𝜋. Now, of course, we can cancel a factor of 𝜋 from the numerator and denominator of this fraction, which leaves us with 𝑟 squared 𝜃 over two or, equivalently, one-half 𝑟 squared 𝜃.

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 𝑟 square 𝜃.