Can You Solve This Geometry Problem For Singapore Students? Finding The Area Of A Claw. Sunday Puzzle

This geometry problem was asked to students in Singapore. What is the area of the shaded region? The figure is composed of squares with side length 4 and quarter circles.

Watch the video for a solution.

Can You Solve This Geometry Problem From Singapore? The Area Of A Claw

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Answer To Geometry Problem From Singapore

I first approached the problem by considering the area of a quarter circle and then subtracting out shapes in order to find the area of the shaded region.

The answer is 32π – 64.

But there is a much more elegant solution! The key is considering the diagonal of the 3×3 grid. The upper region can be rotated about the center to join with the lower region. The resulting shape is a segment of a quarter circle that has a segment of a smaller quarter circle removed.

The area of a segment of a quarter circle is the area of a quarter circle minus the area of an isosceles right triangle. If the radius r, the area is πr²/4 – r²/2.

The larger circular segment has radius 12, and the smaller one has radius 4. So we can calculate the area:

We get to the answer 32π – 64 in a much more direct manner.

Sources

PLSE style circle problem (Mindscope Khoo channel)
https://www.youtube.com/watch?v=yb_JUZFKi-A

Challenging area question – 4 quarter circles and square
http://studyroom.sg/12/

Read the full article here by Mind Your Decisions