Sector Area Calculator
Calculate the area of a circular sector from its radius and central angle. Enter the radius, choose units (deg, rad or grad), and get the sector area, arc length, perimeter and fraction of the full circle.
Find the Area of a Circular Sector from Radius and Angle
Sector Area
Where:
- \(A\) — sector area (square units, e.g. m²)
- \(L\) — arc length of the sector (same unit as the radius)
- \(P\) — perimeter of the sector (two radii plus the arc)
- \(r\) — radius of the circle
- \(\theta\) — central angle in radians
The sector area is simply the fraction of the circle's total area \(\pi r^2\) that the angle \(\theta\) covers: since a full circle corresponds to \(2\pi\) radians, the sector takes up \(\theta / 2\pi\) of it.
If your angle is given in degrees, convert it first: \(\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \pi / 180\). The calculator does this conversion automatically.
Sector Area – Calculation Example
Example: a circular sector has a radius of 5 m and a central angle of 60°. Find its area, arc length and perimeter.
\[ \theta = 60° \cdot \frac{\pi}{180} = \frac{\pi}{3} \approx 1.047 \text{ rad} \]
\[ A = \frac{\theta}{2} r^2 = \frac{1.047}{2} \cdot 5^2 \approx 13.09 \text{ m}^2 \]
\[ L = \theta r = 1.047 \cdot 5 \approx 5.236 \text{ m}, \qquad P = 2 \cdot 5 + 5.236 \approx 15.236 \text{ m} \]
Result: the sector area is about 13.09 m², the arc length is 5.236 m and the perimeter is 15.236 m. The sector covers \(60/360 = 1/6 \approx 0.167\) of the full circle.
How to use the calculator
- Enter the Radius of the circle and pick its unit (m, cm, mm, ft or in). The area will be shown in the matching square unit.
- Enter the Central angle between the two radii and choose its unit: degrees (deg), radians (rad) or gradians (grad).
- Read the results: Sector area (with the full-circle area for comparison), Arc length, Perimeter, Fraction of the circle and a step-by-step breakdown of the calculation.
The radius and the angle must both be greater than zero. Angles above 360° are accepted — the sector then wraps around the centre more than once, and the area grows accordingly.
What is a circular sector?
A circular sector is the region of a circle bounded by two radii and the arc between them — the shape of a pizza slice, a pie chart segment, or a fan. Every sector is defined by exactly two quantities: the radius \(r\) of the circle it belongs to, and the central angle \(\theta\) that sets how wide the "slice" opens at the centre.
The key idea behind the formula is proportionality. A full circle with angle \(2\pi\) radians has area \(\pi r^2\); a sector with angle \(\theta\) covers exactly \(\theta/2\pi\) of that area. Cancelling \(\pi\) gives the compact result \(A = \frac{\theta}{2} r^2\). For a fixed radius, the area grows in direct variation with the angle: double the angle, double the area. Doubling the radius, on the other hand, quadruples the area because \(r\) is squared.
Interpreting the results is straightforward: the arc length tells you how long the curved edge of the sector is, useful for material estimates (trim, fencing, bent tubing). The perimeter adds the two straight radii to the arc — the full boundary length of the slice. The fraction of the circle puts the size in perspective: a 90° sector is a quarter of the circle, a 60° sector a sixth, and so on, which is exactly how pie chart proportions work.
Units, assumptions and tips
- Angle units matter. The formula requires radians. If you work by hand with degrees, convert first: multiply by \(\pi/180\). The calculator converts deg, rad and grad automatically. One gradian is 0.9°, and a full turn is 400 grad.
- Square the length unit for the area. A radius in metres gives an area in m²; a radius in centimetres gives cm². Mixing units (e.g. radius in metres, expecting cm²) is a common source of errors.
- The sector is a plane figure. The calculation assumes a flat circular sector; it does not apply to spherical or conical sectors (such as the curved surface of a cone).
- Angles over 360° produce overlapping sectors that wrap around more than a full turn — the area is then larger than the whole circle. Use them only if that is genuinely what you need.
- Special cases: at \(\theta = 2\pi\) the sector becomes the full circle with area \(\pi r^2\); at \(\theta = \pi\) it is a semicircle with area \(\pi r^2/2\) and perimeter \(P = 2r + \pi r\).
Quick reference table
Fraction of a circle for common angles:
| Central angle | In radians | Fraction of circle |
|---|---|---|
| 30° | \(\pi/6 \approx 0.524\) | 1/12 |
| 45° | \(\pi/4 \approx 0.785\) | 1/8 |
| 60° | \(\pi/3 \approx 1.047\) | 1/6 |
| 90° | \(\pi/2 \approx 1.571\) | 1/4 |
| 180° | \(\pi \approx 3.142\) | 1/2 |
| 270° | \(3\pi/2 \approx 4.712\) | 3/4 |
| 360° | \(2\pi \approx 6.283\) | 1 |
To get the area from this table, simply multiply the fraction by the full-circle area \(\pi r^2\) — or let the calculator do it for you.
Frequently Asked Questions
How do I find the sector area if I only know the arc length (not the angle)?
First find the central angle from the arc length: since \(L = \theta r\), the angle in radians is \(\theta = L / r\). Then plug it into the area formula \(A = \frac{\theta}{2} r^2\), which simplifies nicely to \(A = \frac{L \cdot r}{2}\). For example, a 6 m arc on a radius of 4 m gives \(A = 6 \cdot 4 / 2 = 12\) m². Alternatively, enter the arc length divided by the radius as the angle in radians in the calculator.
What is the difference between a sector and a segment of a circle?
A sector is bounded by two radii and the arc — a pizza slice. A segment is the region between a chord and the arc, without the triangle reaching the centre. If you need the segment area, compute the sector area and subtract the triangle formed by the two radii and the chord: \(A_{\text{segment}} = \frac{\theta}{2} r^2 - \frac{1}{2} r^2 \sin\theta\) (with \(\theta\) in radians). This calculator gives you the sector area, which is the first term of that expression.
Can I use this calculator for a semicircle or a quarter circle?
Yes. A semicircle is just a sector with a 180° angle and a quarter circle is a sector with a 90° angle — enter those values and read the results. The semicircle of radius \(r\) has area \(\pi r^2 / 2\) and perimeter \(2r + \pi r\); the quarter circle has area \(\pi r^2 / 4\) and perimeter \(2r + \pi r / 2\). Even a full circle works: set the angle to 360° and the 'sector' becomes the entire circle.
Why is my hand calculation different from the calculator result?
The most common cause is forgetting to convert degrees to radians. The formulas \(A = \frac{\theta}{2} r^2\) and \(L = \theta r\) only work with \(\theta\) in radians, so 60° must become \(\pi/3 \approx 1.047\) before plugging in. Other frequent mistakes: using the diameter instead of the radius, mixing units (radius in cm but expecting m²), and forgetting that the perimeter includes the two straight radii in addition to the arc.
How do I convert between degrees, radians and gradians?
A full turn is 360° = \(2\pi\) radians = 400 gradians. To convert degrees to radians, multiply by \(\pi/180\) (so 90° → \(\pi/2 \approx 1.571\) rad). To go back, multiply radians by \(180/\pi\). One gradian is 0.9°, so gradians convert to degrees by multiplying by 0.9. The calculator handles all three units automatically, so you only need these conversions when working by hand.
How do I calculate the sector area from a chord length and radius?
First recover the central angle from the chord. The relation is \(c = 2r \sin(\theta/2)\), so \(\theta = 2 \arcsin(c / (2r))\) in radians. Then use \(A = \frac{\theta}{2} r^2\). For instance, a 10 cm chord on a radius of 10 cm gives \(\theta = 2 \arcsin(0.5) = \pi/3\), so \(A = \frac{\pi}{6} \cdot 100 \approx 52.36\) cm². Note that arcsin returns angles up to 180°; for a major sector, use \(2\pi - \theta\) instead.
What units will the results be in, and how do I convert the area?
The arc length and perimeter come out in the same unit as the radius; the area comes in that unit squared — a radius in m gives m², a radius in ft gives ft². To convert the area, multiply by the squared conversion factor: 1 m² = 10 000 cm² = 1 000 000 mm² ≈ 10.764 ft², and 1 ft² = 144 in². If you need a mixed result (say radius in inches, area in ft²), divide the in² result by 144.