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Chord length calculator

For a chord spanning 120° in a circle of radius 8, the chord length calculator gives 8√3, about 13.856: twice the radius times the sine of half the angle.

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.

Chord length8√3 ≈ 13.856406

Central angle The angle between the two radii, at the center.
120° = 2π/3
Arc length s = rθ = 8 × 2π/3 = 16π/3 ≈ 16.755161. In degrees, s = (θ / 360) × 2πr = (120 / 360) × 2π × 8. θ must be in radians in s = rθ.
16π/3 ≈ 16.755161
Sector area A = ½r²θ = ½ × 8² × 2π/3 = 64π/3 ≈ 67.020643. In degrees, (θ / 360) × πr². Equivalently ½ × r × arc length.
64π/3 ≈ 67.020643
Chord length c = 2r sin(θ/2) = 2 × 8 × sin(60°) = 8√3 ≈ 13.856406. The straight line between the arc's ends, always shorter than the arc.
8√3 ≈ 13.856406
Segment area Segment = sector - triangle = ½r²θ - ½r² sin θ = 64π/3 - (16√3) = 64π/3 - 16√3 ≈ 39.30783. 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.
64π/3 - 16√3 ≈ 39.30783
Sagitta (arc height) h = r(1 - cos(θ/2)) = 8(1 - cos 60°) = 4. The height of the arc above the middle of the chord; with the center's distance to the chord it makes up the radius.
4
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.
16 + 16π/3 ≈ 32.755161
Every common angle at radius 8
Central angleArc lengthSector areaChordSegment area
30° (π/6)4.188790216.7551614.14110470.75516082
45° (π/4)6.283185325.1327416.12293492.5053242
60° (π/3)8.377580433.51032285.7975087
90° (π/2)12.56637150.26548211.31370818.265482
120° (2π/3)16.75516167.02064313.85640639.30783
180° (π)25.132741100.5309616100.53096
270° (3π/2)37.699112150.7964511.313708182.79645
360° (2π)50.265482201.061930201.06193
  • 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 chord length formula is c = 2r sin(θ/2), where θ is the central angle. Draw the two radii to the chord's ends and drop a perpendicular from the center to the chord. It cuts both the chord and the angle in half, making two right triangles with hypotenuse r and angle θ/2 at the center, and the half-chord is the side opposite that angle: r sin(θ/2). Double it. For radius 8 and 120°: c = 2 × 8 × sin 60° = 16 × √3/2 = 8√3, about 13.856.

If you know the distance d from the center to the chord instead of the angle, the same right triangle gives c = 2√(r² - d²). That distance and the sagitta (the height of the arc above the chord) add up to the radius. In the example d = r cos(θ/2) = 8 × ½ = 4, so c = 2√(64 - 16) = 2√48 = 8√3 again, and the sagitta is 8 - 4 = 4.

The reverse problem turns up in practice more often than the forward one. On an arch, a curved window or a length of bent pipe you can measure a chord and its sagitta, and what you want is the radius. Then r = (c²/4 + h²) / (2h). For c = 8√3 and h = 4: (48 + 16) / 8 = 8. This calculator works forward from a radius; that formula is how to get the radius first.

The special angles give exact chords. At 60° the chord equals the radius, because the triangle it makes with the two radii is equilateral. At 90° it is r√2, the diagonal of a square. At 120° it is r√3, and at 180° it is the diameter, the longest chord any circle has. Other angles give a decimal only.

Two mistakes cover most wrong answers: using sin θ instead of sin(θ/2), and a calculator in radian mode reading 60 as 60 radians. A chord is always shorter than the arc it spans; at radius 8 and 120° the arc is 16π/3, about 16.755, against the chord's 13.856.

The inscribed angle comes from the same picture. Any angle with its vertex on the major arc and its sides through the chord's ends is half the central angle, 60° here. That is the inscribed angle theorem, and it is why a triangle drawn in a semicircle, with the diameter as one side, always has a right angle opposite it.

Circle Calculator FAQ

What is the formula for chord length?

c = 2r sin(θ/2), where r is the radius and θ the central angle. At radius 8 and 120° that is 16 sin 60° = 8√3, about 13.856.

How do I find a chord from its distance to the center?

c = 2√(r² - d²), where d is the perpendicular distance from the center to the chord. A chord 4 from the center of a radius-8 circle is 2√48 = 8√3.

How do I find the radius from a chord and its height?

With chord c and sagitta h (the height of the arc above the chord), r = (c²/4 + h²) / (2h). A chord of 8√3 with a height of 4 gives (48 + 16) / 8 = 8.

What is the longest chord of a circle?

The diameter, at a central angle of 180°, where sin(θ/2) = 1 and the chord is 2r. Every other chord is shorter, and the closer it is to the center the longer it is.

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