부채꼴 및 활꼴 넓이 계산기
반지름과 중심각으로부터 부채꼴 면적, 활꼴 면적, 호의 길이, 현의 길이를 정확하게 계산합니다.
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.
A circular sector is a pie-wedge bounded by two straight radii from the center and the circular arc. Its boundary perimeter is 2r + L. Its area is proportional to central angle θ.
A circular segment is a region bounded strictly between a straight chord line and the circular arc, without extending to the center. 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.
Geometric Derivation: Segment Area Formula
Proving why segment area equals sector area minus the central triangle:
Common Circle Angles & Proportional Areas
Fraction of total circle area (πr²) for standard central angles from 30° to 270°.
| Angle (Degrees) | Angle (Radians) | Circle Fraction | Sector 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 |
| 180° (Semicircle) | π | 1 / 2 (50.00%) | (πr²) / 2 | 157.08 cm² |
4 Detailed Solved Real-World Examples
Realistic engineering, food portioning, and irrigation problems calculated with complete steps.
Windshield Wiper Sweep Area
A wiper blade of length 60 cm rotates through an angle of 110°:
Horizontal Fuel Tank Liquid Cross-Section
A cylindrical diesel 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 (8 Slices)
Find the exact surface area of one slice from an 8-slice pie:
Common Pitfalls in Partial Circle Calculations
When using 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!
A sector includes the two radii leading to the circle center (pizza slice). A segment is only the outer cap cut off by a straight chord.
If you don't know the angle, find radius using chord width c and arch height h: r = (c² / 8h) + (h / 2).
To find a major segment area (over 180°), calculate the minor segment on the opposite side and subtract it from 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.
Mechanical Wiper Mechanisms
Automotive and marine windshield wiper swept clear-view area compliance under 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.
부채꼴과 활꼴 자주 묻는 질문
원의 면적 공식, 반지름, 지름 및 단위 환산과 관련된 주요 질문과 답변
부채꼴과 활꼴의 차이점은 무엇인가요?
부채꼴(Sector)은 원의 중심과 두 반지름, 호로 둘러싸인 조각입니다. 활꼴(Segment)은 현과 호로 둘러싸인 영역으로 원의 중심을 포함하지 않습니다.
활꼴의 면적은 어떻게 계산하나요?
부채꼴 면적에서 두 반지름과 현이 이루는 삼각형의 면적을 뺍니다: A_seg = (1/2)r²(θ - sin θ) [θ는 라디안 단위].
전문 원형 계산기 도구 탐색
필요한 기하학적 매개변수에 맞춤화된 전문 계산기를 선택하세요
Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
지름으로 원의 넓이 계산기
Calculate circle area directly from diameter using A = (πd²)/4.
둘레로 원의 넓이 계산기
Determine circle area directly from perimeter using A = C²/(4π).
반원 넓이 계산기
Semicircle area, curved arc length, and total perimeter with flat base.
중공원(도넛형 링) 면적 계산기
Annulus, washer, and circular ring surface area between concentric circles.
원의 단면적 계산기
Engineering cross-section for circular shafts, rebar, cables & tensile stress.
원형 파이프 면적 계산기
Internal fluid flow area vs solid pipe wall metal cross-sectional area.
적분을 이용한 원의 넓이 구하기
Calculus derivation of circle area with polar, Cartesian, and shell integrals.