Video: Finding the Area of a Circular Sector given Its Circle’s Radius and Its Arc’s Measure in Degrees

An arc has a measure of 63Β° and a radius of 4. Work out the length of the arc. Give your answer in terms of πœ‹ and in its simplest form. Work out the area of the sector. Give your answer in terms of πœ‹ and in its simplest form

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Video Transcript

An arc has a measure of 63 degrees and a radius of four. Work out the length of the arc. Give your answer in terms of πœ‹ and in its simplest form.

For an arc with a radius π‘Ÿ and the angle πœƒ radians, the arc length is π‘Ÿ multiplied by πœƒ. We have indeed been given the angle and the radius for our sector, but the angles in degrees. So let’s recall how to change from degrees into radians.

Two πœ‹ radians is equivalent to 360 degrees. We can scale down by dividing both sides by 360. And we can see that one degree is equal to two πœ‹ over 360 radians. That’s equivalent to πœ‹ over 180. This means we can convert 63 degrees into radians by multiplying it by πœ‹ over 180.

Before we do the multiplication, though, notice that both 63 and 180 have a common factor of nine. 63 divided by nine is seven. And 180 divided by nine is 20. That means that 63 degrees is equivalent to seven πœ‹ over 20 radians.

We now know the radius of our sector and the angle πœƒ. So we can substitute this into the formula for arc length. π‘Ÿ multiplied by πœƒ is four multiplied by seven πœ‹ over 20. Once again, before we actually perform the calculation, spot that both four and 20 have a common factor of four. One multiplied by seven πœ‹ over five is seven πœ‹ over five. And the length of the arc given is seven πœ‹ over five.

Work out the area of the sector. Give your answer in terms of πœ‹ and in its simplest form.

Once again, when we have a sector with a radius π‘Ÿ and an angle πœƒ radians, the sector area is a half multiplied by π‘Ÿ squared multiplied by πœƒ. We already know the radius of this sector and the angle measured in πœƒ. So we can substitute these into the formula. And it gives us one-half multiplied by four squared multiplied by seven πœ‹ over 20.

We can cross cancel by dividing four squared and 20 by four. We can then also cross cancel by dividing four and two by two. One multiplied by two multiplied by seven πœ‹ is 14πœ‹. And then the denominator of this fraction is five.

The area of the sector is 14πœ‹ over five.

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