Try this beautiful problem from Geometry: Ratio of the area of the star figure to the area of the original circle

## Area of the star and circle – AMC-8, 2012 – Problem 24

A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

- $\frac{1}{\pi}$
- $\frac{4-\pi}{\pi}$
- $\frac{\pi – 1}{\pi}$

**Key Concepts**

Geometry

Circle

Arc

## Check the Answer

But try the problem first…

Answer:$\frac{4-\pi}{\pi}$

AMC-8 (2012) Problem 24

Pre College Mathematics

## Try with Hints

First hint

Clearly the square forms 4-quarter circles around the star figure which is equivalent to one large circle with radius 2.

Can you now finish the problem ……….

Second Hint

find the area of the star figure

can you finish the problem……..

Final Step

Draw a square around the star figure. Then the length of one side of the square be 4(as the diameter of the circle is 4)

Clearly the square forms 4-quarter circles around the star figure which is equivalent to one large circle with radius 2.

The area of the above circle is \(\pi (2)^2 =4\pi\)

and the area of the outer square is \((4)^2=16\)

Thus, the area of the star figure is \(16-4\pi\)

Therefore \(\frac{(the \quad area \quad of \quad the \quad star \quad figure)}{(the \quad area \quad of \quad the \quad original \quad circle )}=\frac{16-4\pi}{4\pi}\)

= \(\frac{4-\pi}{\pi}\)

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