Circle Sector Calculator
Calculate the area, arc length, chord, and perimeter of a circular sector from its radius and central angle, in degrees.
How to use this calculator
- 1Enter the radius of the circle.
- 2Enter the central angle of the sector in degrees.
- 3Read the sector's area and the length of its curved arc.
- 4See the chord across the sector and what fraction of the whole circle it is.
How it works
Circular sector
arc length = r × θ (θ in radians) sector area = ½ × r² × θ chord = 2r × sin(θ / 2) θ radians = degrees × π ÷ 180
A circular sector is the region enclosed by two radii of a circle and the arc between their endpoints — the shape of a slice of pie. Its measurements all follow from the central angle expressed in radians, which is simply the angle as a fraction of a full turn multiplied by 2π. The arc length, the curved outer edge, is the radius times the angle in radians. The sector area is half the radius squared times the angle in radians, which is the same as taking the fraction of the full circle's area that the angle represents. The chord — the straight line joining the two ends of the arc, closing off the slice — is found with basic trigonometry as twice the radius times the sine of half the angle. Using radians rather than degrees is what makes these formulas so compact, since a radian is defined precisely so that arc length equals radius times angle.
Worked example
A 60° sector of a circle with radius 5 covers one-sixth of the circle. Its area is ½ × 5² × (π/3) ≈ 13.09, its arc length is 5 × (π/3) ≈ 5.24, and its chord is 2 × 5 × sin(30°) = 5 — the same as the radius, because a 60° sector's chord and radii form an equilateral triangle.
Circle Sector Calculator: the complete guide
Sectors as fractions of a circle
The simplest way to understand a sector is as a fraction of a whole circle, set by its central angle. A full circle spans 360 degrees, so a sector with a central angle of 90 degrees is exactly one quarter of the circle, a 60-degree sector is one sixth, and so on. Both the area of the sector and the length of its arc are that same fraction of the circle's total area and circumference. This makes sectors easy to reason about: work out the fraction, then apply it to the whole circle's measurements.
This fractional view is why pie charts work so intuitively. Each slice is a sector whose angle is proportional to the share of the whole it represents — a category making up a quarter of the data gets a 90-degree slice. The visual area of each slice is proportional to its value precisely because sector area scales with the central angle. Understanding sectors as scaled-down pieces of a circle connects the geometry directly to one of the most common ways of displaying data.
Why radians make the formulas clean
The sector formulas are strikingly simple — arc length is just radius times angle, and area is half the radius squared times angle — but only when the angle is measured in radians rather than degrees. This is not a coincidence; it is the whole point of radians. A radian is defined as the angle that makes an arc exactly as long as the radius, so 'arc length equals radius times angle' is true by construction. Degrees, an arbitrary division of the circle into 360 parts, would clutter every formula with a conversion factor.
This is why radians are the natural unit of angle in mathematics and physics, even though degrees are more familiar in everyday life. Whenever angles appear in calculus, in the physics of rotation and waves, or in these sector formulas, radians keep the expressions clean and the relationships direct. The calculator accepts the more intuitive degrees for input but converts to radians internally, which is exactly how the mathematics is meant to be done. Seeing why the formulas need radians is a small but genuine insight into why that unit exists at all.
Sectors, segments, and where they appear
It is worth distinguishing a sector from a related shape, the segment, which people sometimes confuse. A sector is bounded by two radii and an arc — the full pie slice, including the pointed centre. A segment is bounded by a chord and an arc — just the outer piece you would get by slicing straight across, without the point. The chord that this calculator reports is exactly the line that separates the segment from the triangular part of the sector, and the segment's area is the sector's area minus that triangle.
Sectors appear far beyond geometry classrooms. Engineers designing gears, cams, and turbine blades work with sectors constantly, since rotating machinery is naturally described by angles swept out. Architects use them for arched windows, fan-shaped rooms, and curved staircases. The sweep of a radar, the coverage of a security camera, the area watered by a rotating sprinkler, and the region lit by a spotlight are all sectors. Anywhere something fans out from a central point through an angle, the sector describes the shape and size of what it covers, which is why these compact formulas earn their keep across engineering, design, and everyday problem-solving.
Frequently asked questions
How do I find the area of a sector?
Multiply half the radius squared by the central angle in radians: area = ½ r² θ. Equivalently, take the fraction of the circle the angle represents (angle ÷ 360) times the full circle's area. A 60° sector of radius 5 has an area of about 13.09.
How do I calculate arc length?
Arc length is the radius times the central angle in radians: L = r θ. Convert degrees to radians by multiplying by π ÷ 180. For a 60° sector of radius 5, that's 5 × (π/3) ≈ 5.24. It's the fraction of the circumference the angle covers.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc — the whole pie slice including the centre point. A segment is bounded by a chord and an arc — just the outer piece cut off by a straight line. The segment's area is the sector's area minus the triangle formed by the two radii and the chord.
Why do the formulas use radians?
Because a radian is defined so that arc length equals radius times angle, making the formulas clean and direct. Degrees would require an extra conversion factor. The calculator takes degrees for convenience but converts to radians internally, which is how the geometry naturally works.