The Leaf Inside a Square — Area Without Calculus
A square of side a. Draw two quarter-circles from opposite corners and a leaf appears in the middle. Its area? No calculus, no memorized formula.
Problem
In a square of side a, draw a quarter-circle of radius a centered at each of two opposite corners. Find the area of the leaf-shaped overlap.
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A square of side a. Draw a quarter-circle of radius a from each of two opposite corners.
Why this approach works
Don't reach for an integral. The trick is the double-counted overlap. Each quarter-circle is πa²/4. Adding the two covers the whole square once, but the central leaf twice. So 2·(πa²/4) = a² + leaf, giving leaf = (π/2 − 1)·a².
Proof
No integral needed: the two quarter-circles cover the square once and the leaf twice, so leaf = (sum of two quarters) − square.
Try it yourself
If a = 10, the leaf is (π/2 − 1)·100 ≈ 57.1 — just over half the square (100).
Related mathematics
Beyond MathVoyage
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