Question Video: Finding the Area of a Circular Sector given Its Circleโ€™s Radius and Its Arcโ€™s Measure in Radians Mathematics

A sector has an arc with a measure of 7๐œ‹/8 radians and a radius of 7. Work out the length of the arc. Give your answer in terms of ๐œ‹ and in its simplest form. Work out the area of the sector. Give your answer in terms of ๐œ‹ and in its simplest form.

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Video Transcript

A sector has an arc with a measure of seven ๐œ‹ over eight radians and a radius of seven. Work out the length of the arc. Give your answer in terms of ๐œ‹ and in its simplest form. There is also a second part where you need to work out the area of the sector. And give your answer in terms of ๐œ‹ and in its simplest form.

So, what Iโ€™ve done is Iโ€™ve drawn a sketch to help us understand what weโ€™ve got in this question. So, weโ€™ve got our sector. Iโ€™ve drawn on the radii, a seven each, because weโ€™re told that the radius is seven. And thereโ€™s also the measure of the angle. And the measure of the angle is seven ๐œ‹ over eight radians. And what we want to work out is the length of the arc. And Iโ€™m gonna call that ๐‘ฅ.

And to help us work out the arc length, we have a formula. And that formula is that the arc length is equal to ๐‘Ÿ๐œƒ. And thatโ€™s where ๐œƒ is the measure of the angle of our sector. And it must be in radians. So, letโ€™s use this to work out the arc length, or ๐‘ฅ, in our problem.

So, when we substitute in our values, weโ€™re gonna get ๐‘ฅ, because thatโ€™s our arc length, is equal to seven, our radius, multiplied by our angle, which is seven ๐œ‹ over eight. Well, that gives us ๐‘ฅ is equal to 49๐œ‹ over eight. And thatโ€™s because seven multiplied by seven is 49. And weโ€™ve got 49๐œ‹ over eight. So, our answer is in terms of ๐œ‹. And itโ€™s also in its simplest form. Because 49 and eight donโ€™t share any factors other than one. So therefore, we can say that the length of the arc is 49๐œ‹ over eight.

Okay, so, now we can move on to the second part of the question. And again, as in the first part of the question, we have a formula to help us. And we have a formula that tells us that the area of a sector is equal to a half ๐‘Ÿ squared ๐œƒ, again, where ๐œƒ is in radians. And then just to remind us of the values that weโ€™ve got for ๐‘Ÿ and ๐œƒ, ๐‘Ÿ is equal to seven, and ๐œƒ is equal to seven ๐œ‹ over eight.

So therefore, the area of the sector is gonna be equal to a half multiplied by seven squared multiplied by seven ๐œ‹ over eight. So, this is gonna be equal to 49 over two. And thatโ€™s cause seven squared is 49. And 49 multiplied by a half is 49 over two multiplied by seven ๐œ‹ over eight. So therefore, this is gonna give us a final answer of 343๐œ‹ over 16.

And we got that because 49 multiplied by seven is 343. And two multiplied by eight is 16. I could have worked it out using the calculator. But if you wanted to work out 49 multiplied by seven using a mental method, you could do 50 multiplied by seven which is 350. And then just subtract seven because itโ€™s 49 multiplied by seven which weโ€™re looking for. So, that wouldโ€™ve given us 343.

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