Quarter Circle Area Formula
The area of a quarter circle is derived from the full circle area formula. Since a quarter circle is exactly one-fourth of a complete circle, you simply divide the full circle area by 4:
Area from Radius
Given the radius r, compute the quarter circle area.
Radius from Area
Given the quarter circle area A, find the radius.
Comparison
How different circle fractions relate.
Understanding the Derivation
A full circle with radius r has area A = πr². Since a quarter circle spans exactly 90 degrees out of 360 degrees, it covers 90/360 = 1/4 of the full circle. Therefore the quarter circle area is (1/4)πr² = πr²/4.
Finding Radius from Area
If you know the area of a quarter circle and need to find the radius, rearrange the formula: A = πr²/4, so r² = 4A/π, and r = √(4A/π). This is useful when you know the area of a rounded corner or a quadrant and need the defining radius.
Practical Examples
- A quarter circle with radius 5 cm has area = π(25)/4 = 19.635 cm²
- A quarter circle with radius 12 m has area = π(144)/4 = 113.097 m²
- If a quarter circle area is 50 sq ft, the radius = √(200/π) = 7.979 ft
Applications
Quarter circle area calculations are used in flooring (rounded room corners), landscaping (curved garden beds), manufacturing (rounded metal pieces), and structural engineering (arch cross-sections).