The arc length of a circular sector is 22 centimetres and the central angle is 77 degrees. Find the area of the sector giving the answer to the nearest square centimetre.
Let’s recall what we know about the arc length and area of a circular sector. For a sector with a radius 𝑟 and an angle 𝜃 radians, its arc length is given by 𝑟𝜃. And its sector area is a half 𝑟 squared 𝜃. We know that the arc length of our circular sector is 22 centimetres. And the measure of its central angle is 77 degrees.
We’re going to use this information to form an equation in terms of 𝑟 using the formula for the arc length. We’ll be able to solve this to find the length of the radius of our circle, which we can then substitute into the formula for sector area.
Remember that we said that 𝜃 needed to be in radians. So we’re going to convert 77 degrees to radians. And to do that, we recall that two 𝜋 radians is equal to 360 degrees. We can then divide through by 360. And that tells us that one degree is equivalent to two 𝜋 over 360 radians. This simplifies to 𝜋 over 180.
And we can now see that, to convert from degrees into radians, we multiply by 𝜋 over 180. And 77 degrees is equal to 77𝜋 over 180 radians. This means our arc length is given by 𝑟 multiplied by 77𝜋 over 180. And since we know the arc length to be 22 centimetres, we can form an equation in terms of 𝑟. To solve this equation, we’ll divide through by 77𝜋 over 180. That’s 16.37 and so on.
Now at this stage, we’re going to choose not to round our answer just yet. Instead, we’ll use the unrounded form in the next step of our calculation. The formula for area of a sector is now one-half multiplied by 16.37 squared multiplied by 77𝜋 over 180. That’s 180.072 and so on.
We were told to round our answer correct to the nearest square centimetre. That’s 180 centimetres squared.