A sector is the slice of a circle bounded by two radii and the arc between them, like a wedge cut from a round pie. This calculator finds the area of that slice from the circle's radius and the central angle, which is useful for geometry work, layout problems, garden plans, material estimates, and any task where only part of a circle matters. Instead of calculating the full circle and reducing it by proportion each time, you can enter the two measurements and get the sector area directly.
Quick Answer
For an angle measured in degrees, the sector area is A = (θ / 360) × πr². Here, r is the radius and θ is the central angle in degrees.
How a Circular Slice Gets Its Area
A complete circle covers 360 degrees and has area πr². A sector occupies a fraction of that circle, and the fraction is set entirely by its central angle. An angle of 90 degrees marks one quarter of the circle, 180 degrees marks one half, and 72 degrees marks one fifth. The sector formula simply multiplies the full area by the same fraction, θ/360. This proportional structure is why doubling the angle doubles the area, while doubling the radius multiplies the area by four: the radius is squared, so its effect grows faster.
Angles may also be given in radians. In that form, the area is A = (1/2)r²θ, with θ stated in radians. The two formulas agree when the angle is converted consistently; 180 degrees equals π radians, and 360 degrees equals 2π radians. The radius must run from the center of the circle to its edge. If a problem supplies the diameter instead, halve it before using the formula. All lengths should use one unit, such as centimeters or inches, and the resulting area is expressed in that unit squared.
The arc and the two straight radii play different roles. The radii determine the size of the circle, while the central angle determines how much of the circumference is included. A long, narrow slice can therefore have less area than a compact slice on a larger circle. Related circle work may call for the Arc Length Calculator when the curved edge itself is the measurement you need.
How to Use the Area of a Sector Calculator
- Identify the circle's radius, not its diameter.
- Measure or find the central angle between the two radii.
- Check whether the angle is stated in degrees or radians and use the form that matches it.
- Enter the radius and angle in the calculator fields.
- Read the result as square units, such as square meters or square inches.
- For a reasonableness check, compare the angle with 360 degrees and estimate the matching fraction of the full circle.
A 72-Degree Slice Worked by Hand
Suppose a circular design has a radius of 10 centimeters and the selected sector has a central angle of 72 degrees. First calculate the full circle: π × 10² = 314.159 square centimeters. The sector fraction is 72/360, which simplifies to 0.2. Multiply the full area by that fraction: 314.159 × 0.2 = 62.832. The sector therefore covers 62.83 square centimeters, rounded to two decimal places.
The same proportional check can be done before calculating. Because 72 degrees is exactly one fifth of 360 degrees, the answer must be one fifth of the circle's area. A result near 63 square centimeters fits; a result larger than 314 square centimeters would signal that the angle, radius, or unit basis was entered incorrectly.
- A radius of 6 units with a 90-degree angle gives 28.27 square units.
- A radius of 12 units with a 45-degree angle gives 56.55 square units.
- A radius of 5 units with a 180-degree angle gives 39.27 square units.
Degree, Radius, and Reflex-Angle Pitfalls
The most frequent error is using the diameter as though it were the radius. That single substitution makes the calculated area four times too large. Another is mixing formulas: if the angle is in degrees, the divisor-360 form is the one to use; if it is in radians, the half-r²θ form applies. Entering 1.57 into the degree formula would treat a right angle as a tiny slice, while entering 90 into the radian formula would produce a sector far beyond one circle.
Be careful with angles greater than 180 degrees. The larger region between the radii is a major sector, while the smaller region is a minor sector. If the given reflex angle is 300 degrees, for example, the formula should use 300 for the major sector; using its 60-degree supplement would calculate the complementary minor slice. Also watch sectors described from a chord or from an arc length alone. A chord does not directly provide the central angle, and an arc length must be connected back to the radius and angle before an area calculation is meaningful.
Rounding deserves attention in practical cutting work. Keep π and intermediate values unrounded until the last step, then round the final area to a sensible precision. Very small rounding differences rarely matter in a homework answer, but repeated sectors cut from sheet material can accumulate visible gaps if each piece is rounded aggressively.
Switching Between Degree and Radian Descriptions
The same sector may be specified in either angular system, so conversion is sometimes the first calculation. To change degrees into radians, multiply by π/180. A central angle of 72 degrees becomes 72π/180, or 2π/5 radians. Substituting that into (1/2)r²θ gives the same area as the degree formula. To change radians into degrees, multiply by 180/π. An angle of 1.2 radians is about 68.75 degrees, not 1.2 degrees, and the distinction changes the slice from a broad sector to a sliver.
An arc length can also identify the angle when the radius is known because arc length s equals rθ when θ is in radians. Dividing s by r gives θ, and the area can then be written as (1/2)rs. This alternate route is useful when a curved boundary has been measured directly. It still depends on a true radius and a central angle formed by two radii; the perimeter of the sector is a separate quantity, consisting of the arc plus both straight sides, and should never be substituted for its area.
Where Sector Areas Show Up
Sector calculations appear in pie charts, irrigation coverage, fan and lighting layouts, circular landscaping, machining, and parts cut from round stock. They also provide a route into related measurements: once the angle and radius are known, the Circumference Calculator can help with the full boundary, and broader geometry exercises are collected in the Mathematics Calculators category. Treat the result as the exact plane area defined by the two radii and arc; thickness, waste, overlap, and edge allowances must be handled separately for a physical build.