What Is The Area? HARD Geometry Problem (Quarter Circles in a Square) — Animated Solution

Can you find the area of the blue, green, and orange regions formed by quarter circles inside a square? This is a classic geometry challenge that has been asked in classrooms around the world — and the challenge is to solve it using ONLY geometry (no trigonometry or calculus)!

📐 THE PROBLEM:
In a square with side length 20, quarter circles are drawn between each pair of corners. Find the area of the three distinct regions created by the overlapping arcs.

🧮 THE SOLUTION:
We set up a system of three equations with three unknowns (a, b, c) by analyzing:
1️⃣ The area of the square: 4a + 4b + c = r²
2️⃣ The area of a quarter circle: 2a + 3b + c = πr²/4
3️⃣ The lens shape formed by two intersecting quarter circles (using an equilateral triangle and circular sectors): a + 2b + c = πr²/3 − r²√3/4

✅ RESULTS (for r = 20):
• Region a (corner piece) ≈ 17.36
• Region b (side lens) ≈ 51.13
• Region c (center rounded square) ≈ 126.06

This animated walkthrough breaks down every step visually so you can follow along easily. Perfect for students preparing for math competitions, SAT/GRE, or anyone who loves geometry puzzles!

Based on the MindYourDecisions video by Presh Talwalkar.

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⏱️ Timestamps:
0:00 — Problem Introduction
0:23 — Setting Up the Regions
0:39 — Equation 1: Area of the Square
0:49 — Equation 2: Area of a Quarter Circle
1:01 — Equation 3: Lens Shape Area
1:28 — Solving the System of Equations
1:36 — Result: Region a
1:49 — Result: Region b
2:02 — Result: Region c
2:15 — Final Summary

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Видео What Is The Area? HARD Geometry Problem (Quarter Circles in a Square) — Animated Solution канала Power Of Pi