Video: Finding the Area Shaded Part between a Circle and a Square

Determine, to the nearest tenth, the area of the shaded region.

02:05

Video Transcript

Determine, to the nearest tenth, the area of the shaded region.

We begin by noticing that this circle is inscribed within a square. So to find the area of the shaded region, we would begin by finding the area of the square and then subtract the area of the circle within it. To find the area of a square, we multiply the length by the length. But what would this length be? Well, as the circle sits exactly within the square, this means that the lengths of all the sides of the square would be the same as the diameter of the circle, which is 19.7 yards. And so our area is 19.7 times 19.7, 388.09. And as it’s an area, our units will be squared, so we’ll have square yards. Then, to find the area of the circle, we recall that this is equal to 𝜋 times the radius squared. We’ll need to be careful when we’re plugging in the values here as the radius is not 19.7 because that’s the diameter.

To find the radius, we half the diameter, so we’ll be calculating 𝜋 times 9.85 squared. We can then put that directly into our calculator, being careful just to square the 9.85 and not the 𝜋 as well, giving us a value of 304.805173 and so on square yards. We won’t round this decimal value yet until we reach the final stage of the question. Putting these together then to find the area of the shaded region, we’ve got the area of our square and the area of our circle. So we have 388.09 subtract 304.805 and so on, which gives us 83.284826 and so on square yards. And since we’re asked to round it to the nearest tenth, that means we check our second decimal digit and see if it’s five or more. And as it is, then our answer rounds up to 83.3 square yards for the area of the shaded region.

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