Try this beautiful problem from Geometry based on the Area of the figure.

## Area of the figure – AMC-8, 2014- Problem 20

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

- $3.5$
- $ 4.0$
- $4.5$

**Key Concepts**

Geometry

Rectangle

Circle

## Check the Answer

But try the problem first…

Answer: $4.0$

AMC-8 (2014) Problem 20

Pre College Mathematics

## Try with Hints

First hint

To Find out the area of the region inside the rectangle but outside all three circles,( i.e the red shaded region in the above figure), we have to find out the area of the Rectangle -the area of three quarter circle inside the circle(i.e green shaded region)

Can you now finish the problem ……….

Second Hint

To find out the area of the rectangle, AD=5 and CD=3 are given. to find out the area of the three-quarter circles, the radii are 1,2 & 3 respectively.

Now area of Rectangle=\(AD \times CD\) and area of quarter circles =\(\frac{\pi r^2}{4}\),where \(r\)=Radius of the circle

can you finish the problem……..

Final Step

Area of the rectangle=\( 5 \times 3\)=15 sq.unit

Area of the quarter circle with the center C=\(\frac{\pi (3)^2}{4}\)=\(\frac{9 \pi}{4}\) sq.unit

Area of the quarter circle with the center B= \(\frac{\pi (2)^2}{4}\) =\(\frac{4 \pi}{4}\)=\(\pi\) sq.unit

Area of the quarter circle with the center C= \(\frac{\pi (1)^2}{4}\) =\(\frac{\pi}{4}\) sq.unit

Therefore the area of the region inside the rectangle but outside all three circles,( i.e the red shaded region in the above figure) =(15- \(\frac{9 \pi}{4}\)- \(\pi\) – \(\frac{\pi}{4}\) )=15-\(\frac{7\pi}{2}\)=15-11=4 sq.unit (Taking \(\pi =\frac{22}{7}\))

## Other useful links

- https://www.cheenta.com/area-of-a-regular-hexagon-amc-8-2012-problem-23/
- https://www.youtube.com/watch?v=LArf0rlglzk

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