The radius of a circle is 28 centimeters, and the arc length of a sector is 37 centimeters. Find the area of the sector.
So we’ve been given two key pieces of information in this question: firstly the radius of a circle and then the arc length of a sector that has been cut from this circle. Let’s draw a sketch of this first. We don’t know the angle at the center of the circle that forms this sector. So I’ve chosen to give it the label 𝜃.
The question asks us to find the area of the sector. So let’s recall the formula for doing so. It is 𝜃, the angle at the center, divided by 360 multiplied by 𝜋𝑟 squared. 𝜋𝑟 squared gives the area of the full circle. And then 𝜃 over 360 scales this according to what portion of the circle we have in this sector.
We know the value of 𝑟. It’s 28 centimeters. But in order to find this area, we need to know the value of 𝜃. Let’s think about the other piece of information we know, the arc length of the sector. Arc length is calculated by finding the circumference of the full circle, two 𝜋𝑟, and then multiplying by 𝜃 over 360, as the arc length only represents a portion of the circumference.
Let’s substitute in the values that we know in this question. The arc length is 37 centimeters, and the radius is 28 centimeters. So we have the equation 𝜃 over 360 multiplied by two multiplied by 𝜋 multiplied by 28 is equal to 37. Now we could solve this equation exactly in order to find the value of 𝜃. But remember, the purpose here is to find the area of the sector. I don’t actually need to know the value of the 𝜃 exactly.
If we compare this equation with the formula for the area of a sector, there are lots of terms that they have in common. They both have 𝜃 over 360, and they both have 𝜋. So actually what I’m going to do is solve not for 𝜃, but 𝜃 over 360 multiplied by 𝜋 and then just substitute this value into my formula for the area of the sector.
So I divide both sides of the equation by two and by 28. And I now have that 𝜃 over 360 multiplied by 𝜋 is equal to 37 over two multiplied by 28. And I’ll leave this as it is for now. Now we can substitute into the formula for the area of the sector. Remember the formula is 𝜃 over 360 multiplied by 𝜋 multiplied by 𝑟 squared. So that first part 𝜃 over 360 multiplied by 𝜋 is 37 over two multiplied by 28 as we’ve just found. And then we multiply it by 𝑟 squared, which is 28 squared.
Now there’s a factor of 28 in the denominator of this fraction and 28 squared in the numerator. So the 28 in the denominator can cancel with one of the 28 squareds in the numerator. So now we have 37 over two multiplied by 28. We can also cancel a factor of two from both the denominator and the numerator. And so now the whole calculation has simplified to 37 multiplied by 14.
Now you could have of course evaluated this on a calculator. But seeing as we haven’t needed a calculator for any other part of the question so far, let’s suppose we don’t have access to one. So you need another method of working out 37 multiplied by 14. You could of course just do a long multiplication.
Or I have divided the 14 up into the sum of 10 and two and two and then worked out 37 times each of those and added them up. It gives a total of 518. So we found then that the area of this sector with its units now is 518 centimeters squared.