Question Video: Circular Sectors and Areas of Circles Mathematics

Video Transcript

Find the area of the shaded part of the quadrant in the diagram in terms of 𝜋.

So we’re looking to find the blue area in the diagram. We’re told that we have a quadrant. That’s a quarter of a circle, which means that this is a sector whose central angle is a right angle. Now, looking carefully at the diagram, we can see that this blue area is made up of a quadrant where the radius of 25 centimeters from which a quadrant of a smaller circle which has a radius of 17 centimeters has been removed. The area we’re looking for then will be the difference between the areas of these two quadrants. Now the question asks us to give our answer in terms of 𝜋, we might be asked to do this for two reasons, either because an exact answer is required or because we’re expected to answer this problem without a calculator.

We haven’t been told whether to work in degrees or radians, so let’s choose to work in degrees. We know that the area of a sector whose central angle is 𝜃 and whose radius is 𝑟 is 𝜃 over 360 multiplied by 𝜋𝑟 squared. In this problem, the central angle is 90 degrees, so we have 90 over 360 multiplied by 𝜋𝑟 squared, which simplifies to one-quarter multiplied by 𝜋𝑟 squared or 𝜋𝑟 squared over four. This, of course, makes sense because we know that a quadrant is quarter of a circle, so the area of a quadrant will be quarter of the full circle’s area. For the area of the larger quadrant first of all then, we have one-quarter multiplied by 𝜋 multiplied by 25 squared. And for the smaller quadrant, it’s one-quarter multiplied by 𝜋 multiplied by 17 squared.

If we want, we can factor by 𝜋 over four giving 𝜋 over four multiplied by 25 squared minus 17 squared. 25 squared is 625 and 17 squared is 289. 625 minus 289 is 336. So we have 336𝜋 over four. And finally, we can cancel a factor of four from the numerator and denominator to give 84𝜋. The units for the radius were centimeters, and so the units for the area will be square centimeters. So we found that the area of the shaded part of the quadrant in terms of 𝜋 is 84𝜋 square centimeters.