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/8 radians and a radius of 7. Work out the length of the arc. Give your answer in terms of ๐ and in its simplest form. Work out the area of the sector. Give your answer in terms of ๐ and in its simplest form.

03:04

Video Transcript

A sector has an arc with a measure of seven ๐ over eight radians and a radius of seven. Work out the length of the arc. Give your answer in terms of ๐ 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 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 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๐. And thatโs where ๐ 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 over eight. Well, that gives us ๐ฅ is equal to 49๐ over eight. And thatโs because seven multiplied by seven is 49. And weโve got 49๐ over eight. So, our answer is in terms of ๐. 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 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 squared ๐, again, where ๐ is in radians. And then just to remind us of the values that weโve got for ๐ and ๐, ๐ is equal to seven, and ๐ is equal to seven ๐ over eight.

So therefore, the area of the sector is gonna be equal to a half multiplied by seven squared multiplied by seven ๐ 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 over eight. So therefore, this is gonna give us a final answer of 343๐ 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.