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Sector area calculator

The sector area calculator opens on a 120° sector of a circle with radius 6: an area of 12π, about 37.699, or a third of the whole circle's 36π.

For the one-measurement mode.

A whole number gives answers exact in π, such as 36π.

For arcs and sectors, and for the equation of a circle. A root such as sqrt(5) or 3√2 stays exact.

In the unit below. A multiple of pi, such as 2pi/3, is read as radians and kept exact.

Which result leads. Every one is still calculated.

For the equation of a circle.

For center and radius from an equation. General form, or (x - 2)^2 + (y + 3)^2 = 25. Type squares as x^2 or x².

For the circle through three points.

Sector area12π ≈ 37.699112

Central angle The angle between the two radii, at the center.
120° = 2π/3
Arc length s = rθ = 6 × 2π/3 = 4π ≈ 12.566371. In degrees, s = (θ / 360) × 2πr = (120 / 360) × 2π × 6. θ must be in radians in s = rθ.
4π ≈ 12.566371
Sector area A = ½r²θ = ½ × 6² × 2π/3 = 12π ≈ 37.699112. In degrees, (θ / 360) × πr². Equivalently ½ × r × arc length.
12π ≈ 37.699112
Chord length c = 2r sin(θ/2) = 2 × 6 × sin(60°) = 6√3 ≈ 10.392305. The straight line between the arc's ends, always shorter than the arc.
6√3 ≈ 10.392305
Segment area Segment = sector - triangle = ½r²θ - ½r² sin θ = 12π - (9√3) = 12π - 9√3 ≈ 22.110655. The triangle is the one made by the two radii and the chord; past 180° its term is negative and the segment is larger than the sector.
12π - 9√3 ≈ 22.110655
Sagitta (arc height) h = r(1 - cos(θ/2)) = 6(1 - cos 60°) = 3. The height of the arc above the middle of the chord; with the center's distance to the chord it makes up the radius.
3
Inscribed angle Any angle with its vertex on the rest of the circle and its sides through the arc's ends is half the central angle (the inscribed angle theorem).
60° = π/3
Sector perimeter 2r + s: the two radii plus the arc.
12 + 4π ≈ 24.566371
Every common angle at radius 6
Central angleArc lengthSector areaChordSegment area
30° (π/6)3.14159279.4247783.10582850.42477796
45° (π/4)4.71238914.1371674.59220121.4092449
60° (π/3)6.283185318.84955663.2610987
90° (π/2)9.42477828.2743348.485281410.274334
120° (2π/3)12.56637137.69911210.39230522.110655
180° (π)18.84955656.5486681256.548668
270° (3π/2)28.27433484.8230028.4852814102.823
360° (2π)37.699112113.097340113.09734
  • Exact forms appear where they exist: arc and sector are exact whenever the angle is a fraction of a turn, and the chord, segment and sagitta when half the angle is a multiple of 30° or 45°. Elsewhere the decimal is the answer.
  • Angles typed as a multiple of pi, such as 2pi/3, are read as radians whatever the unit says.

How the Circle Calculator works

The area of a sector is A = (θ / 360) × πr² with the angle in degrees, or A = ½r²θ with the angle in radians. A sector is a slice of the whole disk, so its area is the same fraction of πr² that its angle is of 360°. For radius 6 and 120°: (120 / 360) × π × 36 = 12π, about 37.699. In radians, 120° is 2π/3, and ½ × 36 × 2π/3 = 12π.

A third form is worth knowing: A = ½rs, where s is the arc length. It says a sector has the area of a triangle with base s and height r. That is what it turns into when you cut it into thin wedges, each nearly a triangle with its tip at the center. In the example the arc is 4π, and ½ × 6 × 4π = 12π. If you know the arc and the radius but not the angle, this is the quickest route, and choosing "arc length" as the angle unit above does exactly that.

The common mistakes are using the diameter as r, which makes the area four times too big, and putting degrees into ½r²θ. The other is confusing a sector with a segment. The sector is the slice bounded by two radii and the arc, a slice of pizza. The segment is the region between the chord and the arc: the slice with its triangle cut away. Segment = sector - triangle, and the triangle's area is ½r² sin θ. Here that is ½ × 36 × sin 120° = 9√3, about 15.588, so the segment is 12π - 9√3, about 22.111.

A sector past 180° is fine. A 270° sector is three quarters of the disk, a pie with one slice gone, and the formulas do not change. Its segment (the region cut off by the chord on the arc's side) is then larger than the sector, because the triangle sits on the other side of the chord and is added rather than removed. The ½r² sin θ term turns negative past 180°, and the same subtraction handles it.

Units square: a sector of a circle measured in centimeters has an area in square centimeters. The sector's perimeter, the length of edge around it, is a different quantity again: 2r + s, which here is 12 + 4π, about 24.566.

Going backwards, the radius from a sector's area and angle is r = √(2A / θ) with θ in radians. A sector of area 12π and angle 2π/3 gives √(24π / (2π/3)) = √36 = 6.

Circle Calculator FAQ

What is the formula for the area of a sector?

A = (θ / 360) × πr² with θ in degrees, or A = ½r²θ with θ in radians. A 120° sector of a circle with radius 6 is 12π, about 37.699.

How do I find the area of a sector from the arc length?

A = ½ × r × s, half the radius times the arc length. A sector with radius 6 and an arc of 4π has area 12π. No angle is needed.

How do I find the radius from the sector area?

Rearrange A = ½r²θ: r = √(2A / θ), with θ in radians. An area of 12π at 2π/3 radians gives r = √36 = 6.

Is a sector the same as a segment?

No. A sector is bounded by two radii and an arc; a segment by a chord and an arc. The segment is the sector minus the triangle formed by the two radii and the chord, whose area is ½r² sin θ.

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