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?

Area of star and 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:$\frac{4-\pi}{\pi}$

Source
Suggested Reading

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

Star and circle

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.

circle

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

circle and square

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

star

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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