Problem 3 · 2015 Math Kangaroo
Easy
Geometry & Measurement
areaarea-decomposition
The grey areas of the square with side length a are bounded by a semi-circle and two quarter-circles respectively. What is their total area?
Show answer
Answer: B — \(\frac{a^2}{2}\)
Show hints
Hint 1 of 2
Compare the curved region added at the top with the curved region removed at the bottom.
Still stuck? Show hint 2 →
Hint 2 of 2
The bottom two quarter-circles together make the same half-disk as the top semicircle, so the curves cancel.
Show solution
Approach: let the equal curved areas cancel to half the square
- The top grey is a semicircle of diameter a (radius a/2), area πa²/8.
- The two bottom quarter-circles (radius a/2) together also have area πa²/8.
- The grey gained at the top equals the white lost at the bottom, so the grey is exactly half the square.
- Total grey area = a²/2.
Mark:
· log in to save