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

# Question Video: Deriving a Formula for Calculating the Area of a Sector When Using Radians Mathematics • First Year of Secondary School

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Write an expression for the area of a sector whose arcβs measure is π radians, knowing that the expression for the area of a sector measuring π degrees is ππΒ²π/360.

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### Video Transcript

Write an expression for the area of a sector whose arcβs measure is π radians, knowing that the expression for the area of a sector measuring π degrees is ππ squared π over 360.

So weβre reminded of the formula we can use to calculate the area of a sector when the central angle is given in degrees. And weβ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 weβ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 ππ squared π 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 π squared π over two or, equivalently, one-half π squared π.

So weβ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; itβs one-half π square π.

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