Question Video: Finding the Radius of a Circular Sector given Its Perimeter and Central Angle in Radians Mathematics

Video Transcript

The perimeter of a circular sector is 67 centimetres and the central angle is 0.31 radians. Find the radius of the sector given the answer to the nearest centimetre.

So, what I’ve done here is a sketch of the circular sector. Let’s now set out all the values that we know. Well, we know that the central angle is 0.31 radians, and we know that the perimeter is 67 centimetres. Okay, great, but we want to work out the radius. And, how we’re going to do that? Well, if we think about the perimeter, the perimeter is gonna be equal to the radius plus the radius plus the curved arc. But, we don’t know the length of this. And, how can we work it out? Well, there’s a formula that we know that can help us. And, that is the arc length is equal to 𝑟𝜃, where 𝜃 is the central angle in radians. It must be in radians. So, this won’t work if we’re using degrees.

So, if we substitute this in, we get the perimeter is equal to 𝑟 plus 𝑟 plus 𝑟𝜃. That’s because 𝑟𝜃 is our arc length. So therefore, if we collect the 𝑟 terms, we can say that 𝑃 is gonna be equal to two 𝑟 plus 𝑟𝜃. And then, if we substitute in the values we know, we’ll get 67, because that’s our perimeter, is equal to two 𝑟 plus 0.31𝑟. That’s because 𝑟𝜃 is 0.31 because that’s our central angle. So therefore, if we add two 𝑟 and 0.31𝑟 , we’re gonna get 2.31𝑟. So, we’ve got 67 is equal to 2.31𝑟.

So then, if we want to find out what 𝑟 is, we’re gonna divide each side of our equation by 2.31. And, that’s because if we divide 2.31𝑟 by 2.31, we’re just left with 𝑟, which is what we’re looking for. And, whatever we do to one side of the equation, we must do to the other side of the equation. And when we do this, we get 𝑟 is equal to 29.0043, et cetera. So, that’s the result of dividing 67 by 2.31.

And, I’ve just flipped to the other way around just so we’ve got the 𝑟 in the left-hand side. Well, have we finished there? Well, not quite because we want the answer to the nearest centimetre. So therefore, we can say that the radius of the sector is going to be equal to 29 centimetres. That’s cause we’ve rounded 29.0043, et cetera, to the nearest centimetre.