The radius of a circle is 21 centimeters, and the angle of a sector is 123 degrees. Find the area of the sector, giving the answer to the nearest square centimeter.
A sector is part of a circle enclosed by two radii and part of the circle circumference, which we call an arc. The sector in this question has a radius of 21 centimeters and a central angle of 123 degrees. That’s the angle where the two radii meet at the center of the circle. We’re asked to find the area of this sector. So let’s recall the formula for doing so. The area of a sector with a radius of 𝑟 units and a central angle of 𝜃 degrees is given by 𝜃 over 360 multiplied by 𝜋𝑟 squared. Practically, we can see where this formula comes from. 𝜋𝑟 squared gives the area of the full circle, and then we multiply this by 𝜃 over 360, which is the fraction of the full circle represented by the sector.
As we know the values of 𝑟, the radius of the circle, and 𝜃, the central angle, we can substitute these into the formula to calculate the area of the sector. It gives 123 over 360 multiplied by 𝜋 multiplied by 21 squared. That simplifies to 54,243 over 360 multiplied by 𝜋. We’re asked to give our answer to the nearest square centimeter, so we need to evaluate this as a decimal. That gives 473.359 continuing. We then need to round this answer to the nearest integer, which is 473. The area of a sector with a radius of 21 centimeters and a central angle of 123 degrees to the nearest square centimeter is 473 square centimeters.