2012 AMC 8 Problem 24
Attempt Problem 24 of the 2012 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
24.
A circle of radius 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?
Answer: A
Small hint:
Rearrange the four circular arcs back inside the original circle.
Big hint:
Compare the star with the square formed by joining the four cut points.
Video solution:
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Written solution:
The area of the original circle is .
Join the four quarter-circle endpoints to form a square. The square has diagonals and , so its area is .
The part inside the circle but outside this square has area . Those four pieces are congruent to the pieces inside the square but outside the star.
Thus the star area is . The desired ratio is . Thus, the answer is A .
Problem 24 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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