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AMC 8 Geometry Math Olympiad USA Math Olympiad

Area of star and circle | AMC-8, 2012|problem 24

Try this beautiful problem from AMC 8, 2012 based on the area of star and circle. You may use sequential hints to solve the problem

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


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

AMC-8 (2012) Problem 24

Pre College Mathematics

Try with Hints


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 ……….

find the area of the star figure

can you finish the problem……..

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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