The diameter of a circle is 50 centimetres and the angle of a sector is 70 degrees. Find the area of the sector giving the answer to the nearest square centimetre.
Let’s recall the formula for the area of a circular sector. It’s a half 𝑟 squared 𝜃, where 𝑟 is the radius of the circle and 𝜃 is the angle of the sector in radians. It’s fairly straightforward to work out the radius of our circle. The radius is half the length of the diameter. So it’s one-half of 50 centimetres. That’s 25. So our circle has a radius of 25 centimetres.
But what about the angle? We are given the angle of the sector to be 70 degrees. But what’s that in radians? Well, two 𝜋 radians is equal to 360 degrees. So we can find the value of one degree by dividing through by 360. Two 𝜋 divided by 360 is equivalent to 𝜋 over 180. So one degree is equal to 𝜋 over 180 radians.
To find the number of radians that are equal to 70 degrees, we’re going to multiply this by 70. And when we do, we see that 70𝜋 over 180 becomes seven 𝜋 over 18. So 70 degrees is equal to seven 𝜋 over 18 radians.
And therefore, we can substitute everything we know into our formula for the area of the sector. It’s one-half multiplied by the radius squared, that’s 25 squared, multiplied by 𝜃 in radians. That’s seven 𝜋 over 18. That gives us a value of 381.790 and so on.
We were told to give our answer correct to the nearest square centimetre. The deciding digit here is the seven. It’s the first digit after the decimal point. Since seven is greater than five, that tells us we round our number up.
And the area of the sector is 382 centimetres squared correct to the nearest square centimetre.