Partial Circle Area Calculator
Calculate the exact area of any circular sector (wedge) or circular segment (chord cap). Supports degrees and radians with instant arc length, chord length, sagitta height, and perimeter solutions.
Partial Circle Area Calculator
Angle subtended at the circle center (0° to 360° or 0 to 2π rad).
What Is a Partial Circle? Sector vs. Segment Explained
In Euclidean geometry, any non-complete circle area falls into two distinct categories: circular sectors and circular segments.
The Pizza Slice
Enclosed by two straight radii running from the center and the circular outer arc. Its boundary perimeter is 2r + L. Its area is directly proportional to the central angle θ.
The Chord Cutout
Enclosed between a straight chord line and the circular arc. It does not touch the circle's center point. Its area equals the sector area minus the triangle between radii and chord.
Partial Circle Dimensional Formulas
| Property | Degrees Formula (θ°) | Radians Formula (θ_rad) | Example (r = 10, θ = 60°) |
|---|---|---|---|
| Arc Length (L) | L = (θ / 360) × 2πr | L = r × θ | 10.47 cm |
| Chord Length (c) | c = 2r × sin(θ / 2) | c = 2r × sin(θ / 2) | 10.00 cm |
| Sagitta / Height (h) | h = r × [1 - cos(θ / 2)] | h = r - √(r² - c²/4) | 1.34 cm |
| Sector Area (A_sec) | A = (θ / 360) × πr² | A = ½ r² θ | 52.36 cm² |
| Central Triangle (A_tri) | A = ½ r² sin(θ) | A = ½ r² sin(θ) | 43.30 cm² |
| Segment Area (A_seg) | A_sec - A_tri | ½ r² (θ - sin θ) | 9.06 cm² |
4 Ways to Calculate Partial Circle Geometry
Choose your method depending on whether your angle is given in degrees, radians, arc length, or chord dimensions.
Degree Sector Method: A = (θ / 360°) × πr²
Divide central angle θ by 360° to find the circle fraction, then multiply by full circle area πr².
Radian Sector Method: A = ½ r² θ
When angle θ is in radians, the formula simplifies elegantly to half of radius squared times angle in radians.
Segment Area: A = ½ r² (θ - sin θ)
Subtract the chord triangle area from the sector. Note that θ inside (θ - sin θ) must be evaluated in radians!
Parabolic Approximation: A ≈ (2/3) c × h
Huygens-Archimedes parabolic approximation for shallow circular segments where chord c and sagitta h are measured directly on site.
Common Circle Angles & Proportional Areas
Fraction of total circle area (πr²) for standard central angles from 15° to 300°.
| Angle (Degrees) | Angle (Radians) | Circle Fraction | Sector Area Formula | Segment Area (r = 10) |
|---|---|---|---|---|
| 30° | π / 6 | 1 / 12 (8.33%) | (πr²) / 12 | 1.18 cm² |
| 45° | π / 4 | 1 / 8 (12.50%) | (πr²) / 8 | 3.94 cm² |
| 60° (Sextant) | π / 3 | 1 / 6 (16.67%) | (πr²) / 6 | 9.06 cm² |
| 90° (Quadrant) | π / 2 | 1 / 4 (25.00%) | (πr²) / 4 | 28.54 cm² |
| 120° | 2π / 3 | 1 / 3 (33.33%) | (πr²) / 3 | 61.42 cm² |
| 135° | 3π / 4 | 3 / 8 (37.50%) | (3πr²) / 8 | 82.46 cm² |
| 180° (Semicircle) | π | 1 / 2 (50.00%) | (πr²) / 2 | 157.08 cm² |
| 240° (Major) | 4π / 3 | 2 / 3 (66.67%) | (2πr²) / 3 | 252.74 cm² |
| 270° (3 Quarters) | 3π / 2 | 3 / 4 (75.00%) | (3πr²) / 4 | 285.62 cm² |
4 Detailed Solved Real-World Examples
Realistic engineering, food portioning, and irrigation problems calculated with complete steps.
Windshield Wiper Sweep Area
A driver's side wiper blade of length 60 cm rotates through an angle of 110°:
Horizontal Fuel Tank Liquid Cross-Section
A cylindrical diesel fuel tank of radius 5 feet is filled to a central angle of 90°:
Center-Pivot Farm Sprinkler Sector
A rotating irrigation arm of length 400 meters waters an arc of 270° around an obstacle:
Single Slice of a 16-Inch Pizza Cut into 8 Slices
Find the exact surface area of one slice from an 8-slice pie:
Common Pitfalls in Partial Circle Calculations
When using the segment equation ½ r² (θ - sin θ), the first θ MUST be in radians. If you enter degrees for θ, your segment area will be completely wrong!
Remember: a sector includes the two radii leading to the circle's center (pie slice). A segment is only the outer cap cut off by a straight line (chord).
If you don't know the angle, you can find the circle radius using just the chord width c and arch height h: r = (c² / 8h) + (h / 2).
To find the area of a major segment (over 180°), simply calculate the minor segment on the opposite side and subtract it from the total circle area πr².
Industry Applications of Sector & Segment Geometry
Civil & Highway Design
Horizontal circular curve sight distances, road median sector easements, and asphalt pave areas.
Industrial Tank Gauging
Dipstick depth calibration for horizontal petroleum, propane, and water tanks using segment geometry.
Agricultural Irrigation
Center-pivot sprinkler boom coverage calculations for partial circular field corners and boundaries.
Mechanical Wiper Mechanisms
Automotive and marine windshield wiper swept clear-view area compliance under federal safety standards.
Architecture & Masonry
Segmental arch windows, curved brick lintels, and Gothic fan vault ceiling masonry load distributions.
Solar & Radar Coverage
Radar sweep azimuth angle detection zones and rooftop solar panel shading arcs across seasons.
Frequently Asked Questions About Sector & Segment Partial Circle Calculations
Answers to the most common questions about calculating the area of a circle, formulas, and units
What is the difference between a circle sector and a circle segment?
A sector is a wedge-shaped region bounded by two radii extending from the center and the circular arc (like a slice cut from the center of a pizza or pie). A segment is the region bounded strictly between a straight chord line and the circular arc, without extending to the center point.
What is the formula for the area of a circular sector?
When central angle θ is in degrees: Area_sector = (θ / 360°) × πr². When central angle θ is in radians: Area_sector = ½ × r² × θ. Both formulas calculate the exact proportional fraction of the full circle's area (πr²).
How do you calculate the area of a circular segment?
A segment's area equals the area of the entire sector minus the area of the central isosceles triangle formed by the two radii and the chord: Area_segment = Area_sector - Area_triangle. In degrees: [(θ / 360°) × πr²] - [½ × r² × sin(θ)]. In radians: ½ × r² × (θ - sin θ).
How do you convert between degrees and radians?
To convert degrees to radians: multiply degrees by (π / 180). For example, 90° × (π / 180) = π / 2 ≈ 1.5708 radians. To convert radians to degrees: multiply radians by (180 / π). For example, 1 radian × (180 / π) ≈ 57.2958°.
What is the chord length formula for a circle segment?
The chord length connecting the two arc endpoints is c = 2r × sin(θ / 2) in trigonometry. If you know the chord length c and radius r, you can solve for the central angle: θ = 2 × arcsin[c / (2r)].
What is the sagitta (height) of a circular segment?
The sagitta (h) is the maximum perpendicular distance from the center of the chord to the arc peak: h = r - r × cos(θ / 2) = r × [1 - cos(θ / 2)]. It is fundamental in civil optics, masonry arches, and horizontal fluid tank depth measurements.
How do you calculate liquid volume in a partially filled horizontal cylinder tank?
When a horizontal cylindrical tank is filled to depth h, the liquid cross-section is a circular segment. Compute the segment area A_segment using depth h and tank radius r, then multiply by tank length L: Volume = A_segment × L.
What is the area of a quadrant (quarter circle)?
A quadrant has a central angle of 90° (π/2 radians). Its sector area is exactly one-quarter of the full circle: A = (πr²) / 4.
What is the area of a sextant (60° sector)?
A sextant has a central angle of 60° (π/3 radians). Its sector area is (60 / 360) × πr² = (πr²) / 6.
What happens if the central angle is greater than 180°?
For θ > 180°, the sector is called a major sector, and the segment is a major segment. For a major segment, the triangular area is added to the sector: Area = ½ r² (θ - sin θ), which naturally holds because sin(θ) is negative for 180° < θ < 360°.
What is the arc length of a partial circle?
In degrees: Arc length L = (θ / 360°) × 2πr = (π × r × θ) / 180°. In radians: Arc length L = r × θ.
How do you find the total perimeter of a circular sector?
A closed sector has three boundary edges: two straight radii and one curved arc. Perimeter_sector = 2r + Arc length = 2r + [(θ / 360°) × 2πr].
Explore Specialized Circle Calculators
Choose a specialized calculation tool tailored to your exact geometric parameters
Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
Area of Circle Calculator With Diameter
Calculate circle area directly from diameter using A = (πd²)/4.
Area of Circle Calculator With Circumference
Determine circle area directly from perimeter using A = C²/(4π).
Half Circle Area Calculator
Semicircle area, curved arc length, and total perimeter with flat base.
Hollow Circle Area Calculator
Annulus, washer, and circular ring surface area between concentric circles.
Cross-Sectional Area of Circle Calculator
Engineering cross-section for circular shafts, rebar, cables & tensile stress.
Circular Pipe Area Calculator
Internal fluid flow area vs solid pipe wall metal cross-sectional area.
Area of Circle Using Integration
Calculus derivation of circle area with polar, Cartesian, and shell integrals.