Калкулатор за Сектор и Сегмент на Кръг
Намерете лицето на сектора, лицето на сегмента, дължината на дъгата и хордата по радиус и централен ъгъл.
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.
Често задавани въпроси: Сектор и сегмент
Отговори на най-важните въпроси за формулите, радиуса, диаметъра и мерните единици
Каква е разликата между сектор и сегмент?
Секторът е ограничен от два радиуса и дъга (включва центъра). Сегментът е ограничен от хорда и дъга (не включва центъра).
Специализирани калкулатори за кръг
Изберете инструмент за изчисление според вашите геометрични параметри
Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
Калкулатор за Лице на Кръг по Диаметър
Calculate circle area directly from diameter using A = (πd²)/4.
Калкулатор за Лице на Кръг по Обиколка
Determine circle area directly from perimeter using A = C²/(4π).
Калкулатор за Лице на Полукръг
Semicircle area, curved arc length, and total perimeter with flat base.
Калкулатор за Лице на Кръгов Пръстен
Annulus, washer, and circular ring surface area between concentric circles.
Калкулатор за Напречно Сечение на Кръг
Engineering cross-section for circular shafts, rebar, cables & tensile stress.
Калкулатор за Тръбно Сечение
Internal fluid flow area vs solid pipe wall metal cross-sectional area.
Лице на Кръг чрез Интегриране
Calculus derivation of circle area with polar, Cartesian, and shell integrals.