Quarter Circle Calculator
Calculate area, perimeter, chord, arc length, and other properties of quarter circles
Calculate Quarter Circle Properties
Enter a positive value for the radius (r)
Example Calculations
Example 1: Radius = 6 units
Quarter Area: A = (π × 6²) / 4 = (π × 36) / 4 = 28.27 units²
Arc Length: L = (π × 6) / 2 = 9.42 units
Chord: c = 6 × √2 = 8.49 units
Perimeter: P = 2(6) + 9.42 = 21.42 units
Example 2: Radius = 10 units
Quarter Area: A = (π × 10²) / 4 = 78.54 units²
Arc Length: L = (π × 10) / 2 = 15.71 units
Chord: c = 10 × √2 = 14.14 units
External Area: A_ext = 10² - 78.54 = 21.46 units²
Example 3: Radius = 5 units
Quarter Area: A = (π × 5²) / 4 = 19.63 units²
Perimeter: P = 2(5) + (π × 5) / 2 = 17.85 units
Centroid: (4×5)/(3π), (4×5)/(3π) = (2.12, 2.12)
Formula Reference
Quick Reference
Common Values
Quick Tips
Quarter circle area is exactly 1/4 of full circle area
Arc length is 1/4 of circle circumference
Chord creates a right triangle with two radii
External area equals square area minus quarter area
Understanding Quarter Circles
What is a Quarter Circle?
A quarter circle is exactly one-fourth (1/4) of a complete circle. It's formed by dividing a circle into four equal parts, where each part has a central angle of 90 degrees (π/2 radians). The quarter circle consists of an arc and two radii that meet at a right angle.
Key Components
- •Arc: The curved portion, 1/4 of the circle's circumference
- •Radii: Two straight lines from center to arc endpoints
- •Chord: Straight line connecting the arc endpoints
- •Central Angle: 90° angle between the two radii
Mathematical Properties
Area Relationships
Quarter area is exactly 1/4 of full circle:
A_quarter = (π × r²) / 4
A_full = π × r²
Arc Length
Arc is 1/4 of circle circumference:
L_arc = (π × r) / 2
C_full = 2π × r
Chord Calculation
Using Pythagorean theorem:
c² = r² + r² = 2r²
c = r × √2
External Area Concept
The external area refers to the area outside the quarter circle but inside a square formed by the two radii and two additional lines perpendicular to them. This creates a square with side length r, and the external area is the difference between this square's area and the quarter circle's area.
External Area = r² - (π × r²) / 4
External Area = r² × (1 - π/4)
Real-World Applications
Architecture & Construction
- • Rounded corner designs in buildings
- • Garden landscaping and pathways
- • Architectural arches and doorways
- • Interior design corner treatments
- • Swimming pool and pond design
Manufacturing & Design
- • Sheet metal cutting and forming
- • Furniture design with rounded corners
- • Automotive part manufacturing
- • Electronic device housing design
- • Textile and clothing patterns
Science & Engineering
- • Mechanical linkage design
- • Flow dynamics in pipe bends
- • Optical lens and mirror design
- • Antenna radiation pattern analysis
- • Stress analysis in curved structures