A sector has an arc with a measure of seven 𝜋 over eight radians and a radius of seven. Work out the length of the arc. Give your answer in terms of 𝜋 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 𝜋 and in its simplest form.
So, what I’ve done is I’ve drawn a sketch to help us understand what we’ve got in this question. So, we’ve got our sector. I’ve drawn on the radii, a seven each, because we’re told that the radius is seven. And there’s also the measure of the angle. And the measure of the angle is seven 𝜋 over eight radians. And what we want to work out is the length of the arc. And I’m gonna call that 𝑥.
And to help us work out the arc length, we have a formula. And that formula is that the arc length is equal to 𝑟𝜃. And that’s where 𝜃 is the measure of the angle of our sector. And it must be in radians. So, let’s use this to work out the arc length, or 𝑥, in our problem.
So, when we substitute in our values, we’re gonna get 𝑥, because that’s our arc length, is equal to seven, our radius, multiplied by our angle, which is seven 𝜋 over eight. Well, that gives us 𝑥 is equal to 49𝜋 over eight. And that’s because seven multiplied by seven is 49. And we’ve got 49𝜋 over eight. So, our answer is in terms of 𝜋. And it’s also in its simplest form. Because 49 and eight don’t share any factors other than one. So therefore, we can say that the length of the arc is 49𝜋 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 𝑟 squared 𝜃, again, where 𝜃 is in radians. And then just to remind us of the values that we’ve got for 𝑟 and 𝜃, 𝑟 is equal to seven, and 𝜃 is equal to seven 𝜋 over eight.
So therefore, the area of the sector is gonna be equal to a half multiplied by seven squared multiplied by seven 𝜋 over eight. So, this is gonna be equal to 49 over two. And that’s cause seven squared is 49. And 49 multiplied by a half is 49 over two multiplied by seven 𝜋 over eight. So therefore, this is gonna give us a final answer of 343𝜋 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 it’s 49 multiplied by seven which we’re looking for. So, that would’ve given us 343.