What is the formula for the area of a sector of a circle?

When the central angle θ is in degrees: A = (θ / 360°) * pi * r^2. When the central angle α is in radians: A = 1/2 * r^2 * α. When given radius r and arc length L: A = 1/2 * r * L.

How do you find the area of a shaded sector of a circle?

Identify the circle's radius r and the shaded central angle θ (in degrees). Substitute into A = (θ / 360°) * pi * r^2. For example, a 60° shaded sector with radius r = 6 cm has area A = (60 / 360) * pi * 6^2 = 6*pi ≈ 18.85 cm^2.

What is the formula for the perimeter of a sector?

The perimeter of a closed sector includes the curved arc length plus two straight radii: P = L_arc + 2r = (pi * r * θ) / 180° + 2r = r * ((pi * θ) / 180° + 2) (or P = r * (α + 2) in radians).

How do you calculate arc length from central angle and radius?

For angle θ in degrees: L = (θ / 360°) * 2*pi*r = (pi * r * θ) / 180°. For angle α in radians: L = r * α.

How do you find the central angle of a sector from its area and radius?

Rearranging the sector area formula yields: in degrees, θ = (360° * A) / (pi * r^2); in radians, α = (2 * A) / r^2.

What is the difference between a minor sector and a major sector?

A minor sector has a central angle less than 180° (θ 180°), with an area equal to A_major = pi * r^2 - A_minor.

What is the difference between a circular sector and a circular segment?

A circular sector is a pie-shaped slice bounded by two radii and an arc (A_sector = 1/2 * r^2 * α). A circular segment is the region bounded between a chord and the arc, with area A_segment = A_sector - A_triangle = 1/2 * r^2 * (α - sin(α)).

What is the chord length formula for a circular sector?

The straight chord connecting the two ends of the sector's arc is c = 2r * sin(θ / 2) = 2r * sin(α / 2).

How do you find the centroid of a circular sector?

The centroid of a circular sector with central angle α (in radians) lies on the axis of symmetry at a distance x̄ = (2r * sin(α / 2)) / (3 * (α / 2)) = (4r * sin(θ / 2)) / (3 * α_rad) from the circle center.

How does this Sector Area Calculator work and what input combinations does it support?

The calculator supports multidirectional solving across 5 combinations: (1) Radius & Central Angle (θ° or rad); (2) Radius & Arc Length; (3) Arc Length & Central Angle; (4) Sector Area & Radius; and (5) Sector Area & Central Angle, computing arc length, perimeter, chord length, segment area, and centroid distance.

100% Free • Circular Sector, Arc Length & Perimeter Solver

Sector Area Calculator

Calculate the area of a sector of a circle ($A = \frac{\theta}{360^\circ}\pi r^2 = \frac{1}{2}r^2\alpha = \frac{1}{2}rL$), arc length ($L = \frac{\pi r\theta}{180^\circ}$), closed perimeter ($P = L + 2r$), chord length ($c = 2r\sin(\theta/2)$), circular segment area, and sector centroid distance across degrees and radians.

Sector Presets: Tap to load

Sector Parameters

Select input combination to calculate sector area, perimeter, and arc geometry.

A = (θ/360°)πr² = ½rL

Given Parameters

Angle Measurement Unit

Radius (r)

Central Angle (θ)

Key Dimension Glance

Sector Area (A) 78.540 cm²

Arc Length (L) 15.708 cm

Closed Perimeter 35.708 cm

Circle Fraction 25.00%

Sector Area (A)

78.540 cm²

Radius: 10.000 cm • Angle: 90.000° (1.571 rad) • Arc: 15.708 cm

Geometric Telemetry Matrix

📏 Circle Radius ($r$)	10.000 cm
📏 Circle Diameter ($d$)	20.000 cm
📐 Central Angle ($\theta$)	90.000° (1.571 rad)
🔄 Curved Arc Length ($L$)	15.708 cm
📏 Closed Perimeter ($P$)	35.708 cm (L + 2r)
🔵 Sector Area ($A_{\text{sector}}$)	78.540 cm²
🍕 Circle Coverage Fraction	25.000% (1/4 of circle)
📐 Chord Length ($c$)	14.142 cm (2r sin(θ/2))
📏 Sagitta Height ($h$)	2.929 cm
🔺 Triangle Area under Chord	50.000 cm² (½r² sin θ)
🌘 Circular Segment Area	28.540 cm² (A_sec - A_tri)
🎯 Centroid Distance ($\bar{x}$)	6.002 cm (from center)

Why Use Our Sector Area Calculator? Real-World Problems It Solves

Circular sectors (pie-shaped portions of a circle) are integral to civil highway curve design, agricultural center-pivot irrigation coverage, wind turbine swept areas, architectural arches, and culinary portion sizing. Our Sector Area Calculator solves critical real-world challenges:

1. Agricultural Sprinkler Irrigation Zones:

Calculates exact field acreage watered by rotating sprinkler heads covering partial arcs ($90^\circ$, $180^\circ$, or $270^\circ$) using $A = \left(\frac{\theta}{360^\circ}\right)\pi r^2$.

2. Civil Engineering Highway Curves:

Computes arc length ($L = \frac{\pi r\theta}{180^\circ}$), chord length ($c = 2r\sin(\theta/2)$), and sightline sagitta clearances for horizontal road curves and railway tracks.

3. Circular Segment & Triangle Decomposition:

Decomposes any circular sector into its inner triangle area ($\frac{1}{2}r^2\sin\theta$) and outer circular segment region ($A_{\text{seg}} = A_{\text{sec}} - A_{\text{tri}}$).

4. Multidirectional Inverse Solving:

Solves for radius ($r$), central angle ($\theta$), arc length ($L$), or sector area ($A$) from any pair of given dimensions with seamless degree-to-radian conversions.

How to Calculate the Area and Perimeter of a Circular Sector (Step-by-Step Guide)

Step 1: Choose Given Inputs

Select from Radius & Angle, Radius & Arc Length, Arc & Angle, Area & Radius, or Area & Angle.

Step 2: Choose Angle Unit

Toggle between Degrees (°) and Radians (rad), and enter your numerical values.

Step 3: Review Sector Geometry

Inspect calculated sector area, arc length, perimeter, chord, and segment dimensions.

Step 4: Copy Full Telemetry

Click Copy Sector Telemetry Card to export formatted results for engineering, CAD, or homework.

All Circular Sector Formulas & Equations

1. Central Angle in Degrees ($\theta$)

$$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$$ $$L_{\text{arc}} = \frac{\theta}{360^\circ} \times 2\pi r = \frac{\pi r \theta}{180^\circ}$$ $$P_{\text{sector}} = L_{\text{arc}} + 2r = r\left(\frac{\pi \theta}{180^\circ} + 2\right)$$

2. Central Angle in Radians ($\alpha$)

$$A_{\text{sector}} = \frac{1}{2} r^2 \alpha = \frac{1}{2} r L_{\text{arc}}$$ $$L_{\text{arc}} = r \alpha, \quad P = r(\alpha + 2)$$ $$\text{Relation: } A = \frac{L_{\text{arc}}^2}{2\alpha}$$

3. Chord, Sagitta & Segment Formulas

$$c = 2r \sin\left(\frac{\theta}{2}\right), \quad h_{\text{sagitta}} = r\left(1 - \cos\left(\frac{\theta}{2}\right)\right)$$ $$A_{\text{triangle}} = \frac{1}{2}r^2 \sin(\theta)$$ $$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2}r^2(\alpha - \sin\alpha)$$

4. Sector Centroid Distance ($\bar{x}$)

$$\bar{x} = \frac{2r \sin(\alpha/2)}{3(\alpha/2)} = \frac{4r \sin(\theta/2)}{3\alpha_{\text{rad}}}$$ $$\text{Quadrant } (90^\circ): \bar{x} = \frac{4\sqrt{2}r}{3\pi} \approx 0.6002 \cdot r$$

Common Mistakes in Sector Calculations & How to Avoid Them

Mistake 1: Confusing Arc Length with Sector Perimeter

Arc length ($L$) only measures the curved outer edge of the sector. A closed sector includes the two straight radii borders: $P = L + 2r$.

Mistake 2: Mixing Up Degree and Radian Formulas

Using $\frac{1}{2}r^2\theta$ with $\theta$ in degrees gives wildly incorrect answers! The formula $\frac{1}{2}r^2\alpha$ requires angle $\alpha$ in radians. For degrees, you must use $\frac{\theta}{360^\circ}\pi r^2$.

Mistake 3: Confusing a Sector with a Segment

A circular sector is a pie slice connected to the circle center. A circular segment is only the outer slice between the chord and the arc. Segment Area = Sector Area − Triangle Area.

Mistake 4: Minor vs Major Sector Angles

If asked for the area of a shaded major sector with acute angle $\theta$, the major sector angle is $360^\circ - \theta$. Always ensure you are calculating the intended shaded region.

Classroom Practice & Study Guide: Sector Area Problems

Worked Practice Problems with Solutions:

Problem 1: Finding Sector Area & Perimeter from Radius and Angle

A circular sector has a radius of $r = 12.0\,\text{cm}$ and a central angle of $\theta = 60^\circ$. Calculate its area, arc length, and closed perimeter.

Solution: Sector Area: $A = \frac{60}{360} \times \pi \times 12^2 = \frac{1}{6} \times 144\pi = 24\pi \approx \mathbf{75.398\,\text{cm}^2}$. Arc Length: $L = \frac{60}{360} \times 2\pi(12) = 4\pi \approx \mathbf{12.566\,\text{cm}}$. Closed Perimeter: $P = 12.566 + 2(12) = \mathbf{36.566\,\text{cm}}$.

Problem 2: Finding Central Angle from Known Sector Area

A circle of radius $r = 8\,\text{m}$ has a shaded sector with area $A = 50\,\text{m}^2$. What is the central angle in degrees and radians?

Solution: In radians: $\alpha = \frac{2A}{r^2} = \frac{100}{64} = \mathbf{1.5625\,\text{rad}}$. In degrees: $\theta = 1.5625 \times \frac{180^\circ}{\pi} \approx \mathbf{89.525^\circ}$.

Problem 3: Calculating Circular Segment Area

Find the area of the circular segment formed by a $90^\circ$ sector of radius $r = 10\,\text{cm}$.

Solution: Sector Area: $A_{\text{sector}} = \frac{90}{360}\pi(10^2) = 25\pi \approx \mathbf{78.540\,\text{cm}^2}$. Triangle Area: $A_{\text{tri}} = \frac{1}{2}(10)(10)\sin(90^\circ) = \mathbf{50.000\,\text{cm}^2}$. Segment Area: $A_{\text{seg}} = 78.540 - 50 = \mathbf{28.540\,\text{cm}^2}$.

Master Benchmark Sector Geometry Matrix Table

Standard dimensions across canonical central angles for a circle of radius $r = 10\,\text{cm}$.

Sector Benchmarks

Central Angle ($\theta$)	Radians ($\alpha$)	Sector Area ($r=10$)	Arc Length ($L$)	Chord Length ($c$)	Circle Fraction
$30^\circ$ (1/12th Circle)	$0.524\,\text{rad}$ ($\frac{\pi}{6}$)	$26.180\,\text{cm}^2$	$5.236\,\text{cm}$	$5.176\,\text{cm}$	$8.33\%$
$45^\circ$ (Octant / 1/8th)	$0.785\,\text{rad}$ ($\frac{\pi}{4}$)	$39.270\,\text{cm}^2$	$7.854\,\text{cm}$	$7.654\,\text{cm}$	$12.50\%$
$60^\circ$ (Sextant / 1/6th)	$1.047\,\text{rad}$ ($\frac{\pi}{3}$)	$52.360\,\text{cm}^2$	$10.472\,\text{cm}$	$10.000\,\text{cm}$	$16.67\%$
$90^\circ$ (Quadrant / 1/4th)	$1.571\,\text{rad}$ ($\frac{\pi}{2}$)	$78.540\,\text{cm}^2$	$15.708\,\text{cm}$	$14.142\,\text{cm}$	$25.00\%$
$180^\circ$ (Semicircle / 1/2)	$3.142\,\text{rad}$ ($\pi$)	$157.080\,\text{cm}^2$	$31.416\,\text{cm}$	$20.000\,\text{cm}$	$50.00\%$
$270^\circ$ (Major 3/4th)	$4.712\,\text{rad}$ ($\frac{3\pi}{2}$)	$235.619\,\text{cm}^2$	$47.124\,\text{cm}$	$14.142\,\text{cm}$	$75.00\%$

Real-World Engineering, Agriculture & Design Applications

Everyday Engineering Implementations:

Center-Pivot Crop Irrigation: Calculating acreage irrigated by partial-swing water pivots on agricultural land.
Civil Road Design & Sightlines: Determining the curved distance ($L$) and lateral stopping sight distance (sagitta $h$) on highway curves.
Wind Turbine & Propeller Swept Area: Modeling airflow capture through partial rotor blade sectors.
Rotary Mechanical Cams & Gears: Sizing partial gear teeth arcs and sector-shaped counterweights.

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about circular sector areas, arc lengths, perimeters, chord lengths, and segment calculations.

Central Angle (\(\theta\))	Radians (\(\alpha\))	Sector Area (\(r=10\))	Arc Length (\(L\))	Chord Length (\(c\))	Circle Fraction
\(30^\circ\) (1/12th Circle)	\(0.524\,\text{rad}\) (\(\frac{\pi}{6}\))	\(26.180\,\text{cm}^2\)	\(5.236\,\text{cm}\)	\(5.176\,\text{cm}\)	\(8.33\%\)
\(45^\circ\) (Octant / 1/8th)	\(0.785\,\text{rad}\) (\(\frac{\pi}{4}\))	\(39.270\,\text{cm}^2\)	\(7.854\,\text{cm}\)	\(7.654\,\text{cm}\)	\(12.50\%\)
\(60^\circ\) (Sextant / 1/6th)	\(1.047\,\text{rad}\) (\(\frac{\pi}{3}\))	\(52.360\,\text{cm}^2\)	\(10.472\,\text{cm}\)	\(10.000\,\text{cm}\)	\(16.67\%\)
\(90^\circ\) (Quadrant / 1/4th)	\(1.571\,\text{rad}\) (\(\frac{\pi}{2}\))	\(78.540\,\text{cm}^2\)	\(15.708\,\text{cm}\)	\(14.142\,\text{cm}\)	\(25.00\%\)
\(180^\circ\) (Semicircle / 1/2)	\(3.142\,\text{rad}\) (\(\pi\))	\(157.080\,\text{cm}^2\)	\(31.416\,\text{cm}\)	\(20.000\,\text{cm}\)	\(50.00\%\)
\(270^\circ\) (Major 3/4th)	\(4.712\,\text{rad}\) (\(\frac{3\pi}{2}\))	\(235.619\,\text{cm}^2\)	\(47.124\,\text{cm}\)	\(14.142\,\text{cm}\)	\(75.00\%\)

📏 Circle Radius (\(r\))	10.000 cm
📏 Circle Diameter (\(d\))	20.000 cm
📐 Central Angle (\(\theta\))	90.000° (1.571 rad)
🔄 Curved Arc Length (\(L\))	15.708 cm
📏 Closed Perimeter (\(P\))	35.708 cm (L + 2r)
🔵 Sector Area (\(A_{\text{sector}}\))	78.540 cm²
🍕 Circle Coverage Fraction	25.000% (1/4 of circle)
📐 Chord Length (\(c\))	14.142 cm (2r sin(θ/2))
📏 Sagitta Height (\(h\))	2.929 cm
🔺 Triangle Area under Chord	50.000 cm² (½r² sin θ)
🌘 Circular Segment Area	28.540 cm² (A_sec - A_tri)
🎯 Centroid Distance (\(\bar{x}\))	6.002 cm (from center)