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## 7th grade (Illustrative Mathematics)

### Unit 3: Lesson 6

Lesson 9: Applying area to circles# Area of a shaded region

CCSS.Math:

Here's a fun one: find the area of a shaded region where you first determine the area of a square and then the area of a circle. Created by Sal Khan.

## Video transcript

We're asked to find the area
of the shaded region, so the area of this
red-shaded region. So this is interesting. This is almost a
10 by 10 square, except we have these quarter
circles that are cut out. So the area of this would be the
area of what a 10 by 10 square would be minus the area
of these quarter circles. And each of these
quarter circles is a quarter of a
circle with a radius 3. I think we can assume
that all of these, if you took the
distance from here to the outside of this
quarter circle, have radius 3. So if you put four
quarter circles together, you're going to have a
complete white circle. So one way to
think about this is that the area of
this whole red region is going to be the area of
the entire square, which is 10 by 10. So it's going to be
10 times 10, which is 100 whatever
square units we have. And then we're going
to subtract out the area of the four
quarter circles. And that area is going to be
equivalent to the area of one circle with a radius of 3. So what's the area of
a circle with radius 3? Well, the formula
for area of a circle is pi r squared,
or r squared pi. So the radius is 3. So it's going to be 3 times 3,
which is 9, times pi-- 9 pi. So we have 100 minus 9 pi is
the area of the shaded region. And we got it right.