Quarter Circle Area Calculator
Calculate area, perimeter, chord, arc length, and all properties of quarter circles
Calculate Quarter Circle Properties
Enter a positive value for the radius (r)
Example Calculations
Example 1: Radius = 2 units
Quarter Area: A = (π × 2²) / 4 = (π × 4) / 4 = π ≈ 3.14 units²
Arc Length: L = (π × 2) / 2 = π ≈ 3.14 units
Chord: c = 2 × √2 ≈ 2.83 units
Perimeter: P = 2(2) + π = 4 + π ≈ 7.14 units
Example 2: Radius = 5 units
Quarter Area: A = (π × 5²) / 4 = 25π/4 ≈ 19.63 units²
Arc Length: L = (π × 5) / 2 = 5π/2 ≈ 7.85 units
Chord: c = 5 × √2 ≈ 7.07 units
Spare Area: A_spare = 5² - 19.63 = 5.37 units²
Example 3: Radius = 10 units
Quarter Area: A = (π × 10²) / 4 = 25π ≈ 78.54 units²
Perimeter: P = 2(10) + 5π = 20 + 15.71 = 35.71 units
Full Circle Area: A_full = π × 10² = 100π ≈ 314.16 units²
Formula Reference
Circle Relationships
Common Values
Quick Tips
Quarter circle area is exactly π×r²/4
Arc length is 1/4 of the full circumference
Chord creates a right triangle with radii
Spare area is the remaining square area
Understanding Quarter Circle Area
What is Quarter Circle Area?
The quarter circle area represents exactly one-fourth (1/4) of a complete circle's area. Since a full circle has an area of π×r², a quarter circle has an area of (π×r²)/4. This calculation is fundamental in geometry and has numerous practical applications.
Key Components
- •Quarter Area: (π × r²) / 4 - The main curved region
- •Arc Length: (π × r) / 2 - The curved boundary
- •Chord: r × √2 - Straight line across the arc
- •Spare Area: r² - (π × r²) / 4 - Area outside quarter
Mathematical Derivation
Area Formula Derivation
Starting from full circle area:
A_full = π × r²
A_quarter = A_full / 4
A_quarter = (π × r²) / 4
Arc Length Derivation
From full circumference:
C_full = 2π × r
L_arc = C_full / 4
L_arc = (π × r) / 2
Chord Calculation
Using Pythagorean theorem:
c² = r² + r²
c = √(2r²) = r√2
Understanding Spare Area
The spare area (also called external area) refers to the area within a square of side length r that lies outside the quarter circle. When a quarter circle is inscribed in a square formed by two radii and perpendicular lines, the spare area is the difference between the square's area and the quarter circle's area.
Spare Area = r² - (π × r²) / 4
Spare Area = r² × (1 - π/4)
Since π/4 ≈ 0.785, the spare area is about 21.5% of the square
Real-World Applications
Architecture & Construction
- • Calculating area of rounded building corners
- • Garden design with curved pathways
- • Arch construction and material estimation
- • Interior design for curved walls
- • Pool and water feature design
Manufacturing & Engineering
- • Sheet metal cutting calculations
- • Automotive part design and area calculation
- • Furniture with rounded corners
- • Electronics housing design
- • Textile pattern calculations
Mathematics & Science
- • Geometric problem solving
- • Physics calculations for rotational motion
- • Engineering stress analysis
- • Calculus integration problems
- • Computer graphics and animation
Frequently Asked Questions
How many quarter circles equal a whole circle?
Exactly 4 quarter circles equal a whole circle. That's why to compute the area of a quarter circle, you divide the area of a whole circle by 4: Quarter circle area = Full circle area / 4.
How to calculate quarter circle area with radius = 2?
Using the formula A = (π × r²) / 4 with r = 2:
A = (π × 2²) / 4
A = (π × 4) / 4
A = π ≈ 3.14159 units²
What is the relationship between chord and radius?
In a quarter circle, the chord connects the two endpoints of the arc, forming the hypotenuse of a right triangle with two radii as legs. Using the Pythagorean theorem: chord = r × √2 ≈ 1.414 × r.
How is spare area calculated?
Spare area is the area within a square (formed by extending the radii) that lies outside the quarter circle. It's calculated as: Spare area = r² - (π × r²) / 4 = r² × (1 - π/4).