What is the difference between arc length and chord?

The arc length is the curved distance along the circle's edge; the chord is the straight line between the same two endpoints. The arc is always longer, and the gap grows as the angle widens.

Circle Sector & Arc Calculator

Q: How do you find the arc length of a sector?

With the angle in radians, arc length is radius times angle: L equals r theta. If the angle is in degrees, use L equals (theta over 360) times 2 pi r. A 60 degree arc on a radius of 12 is about 12.57.

Q: How do you find the area of a sector?

In radians the sector area is one half r squared theta. In degrees it is the fraction of the circle times its area: (theta over 360) times pi r squared. For r equals 12 and theta 60 degrees that is about 75.40.

Q: What is the difference between a sector and a segment?

A sector is the full wedge bounded by two radii and the arc, and it includes the centre. A segment is only the region between the chord and the arc, the sector with its triangle removed. The segment is the smaller of the two.

Q: How do you find the radius from the arc length and angle?

Rearrange L equals r theta to r equals L divided by theta, with the angle in radians. A 30 degree arc measuring 10 units long comes from a radius of about 19.1.

A sector is a slice of a circle — two radii and the arc between them. Enter a radius and an angle, or any two of radius, angle, arc length and sector area, and the rest follow, drawn to scale. You can also click a value on the drawing to change it.

Known values

from radius + angle

Enter any two values — the rest update automatically.

Radius (r)

Angle (θ)

Arc length

Sector area

Angle Units Decimals

Click an amber value to edit it · angle drawn true to scale

Results

Sector formulas

A sector is fixed by a radius and a central angle. With the angle in radians, the arc and area are short products; the chord uses half the angle.

Arc · L = r θ

Sector area · A = ½ r² θ

Chord · c = 2 r sin(θ / 2)

Degrees or radians

Those formulas use radians. In degrees, the arc is (θ / 360) × 2πr and the area is (θ / 360) × πr². This calculator takes the angle in degrees and converts for you.

Sector vs segment

A sector is the whole wedge, including the centre. A segment is just the part between the chord and the arc — the sector with its triangle removed.

Working with sectors and arcs

A sector is a slice of a circle bounded by two radii and the arc between their ends — a pizza slice, a pie chart wedge, the sweep of a windscreen wiper. It's set by two things: how far it reaches (the radius) and how wide it opens (the central angle). Give this calculator any two of radius, angle, arc length and sector area, and it recovers the circle the slice came from, then reports every related measurement.

The parts of a sector

Radius (r) — the distance from the centre to the arc, the length of each straight edge of the slice.
Central angle (θ) — how wide the slice opens at the centre, entered here in degrees.
Arc length (L) — the curved outer edge.
Chord — the straight line joining the two ends of the arc, cutting across the opening.
Segment — the smaller region between the chord and the arc; the sector minus the triangle formed by the two radii and the chord.

Worked example

Take a sector with radius r = 12 and central angle θ = 60°. First convert the angle to radians, since the core formulas use radians: 60° × π/180 = 1.0472 rad.

Step 1 — arc length

L = r θ = 12 × 1.0472 = 12.566

Step 2 — sector area

A = ½ r² θ = ½ × 144 × 1.0472 = 75.40

Step 3 — chord length

c = 2r sin(θ/2) = 24 × sin 30° = 24 × 0.5 = 12

Step 4 — fraction of the whole circle

θ / 360° = 60 / 360 = 1/6 of the full circle

Result

Arc 12.566, sector area 75.40, chord 12. At 60° the chord equals the radius, so this slice is exactly one-sixth of the circle — the shape you'd get cutting a round pizza into six.

Formulas this calculator uses

Arc length · L = r θ

Sector area · A = ½ r² θ

Chord · c = 2 r sin(θ / 2)

Segment area · ½ r² (θ − sin θ)

Arc height (sagitta) · r (1 − cos(θ / 2))

Angles are entered in degrees and converted to radians (θ × π / 180) for the calculation. The perimeter of a sector is the arc plus the two radii: L + 2r.

Frequently asked questions

How do you find the arc length of a sector?

With the angle in radians, arc length is simply radius times angle: L = r θ. If your angle is in degrees, use L = (θ / 360) × 2πr instead. A 60° arc on a radius of 12 is about 12.57.

How do you find the area of a sector?

In radians, the sector area is A = ½ r² θ. In degrees it's the fraction of the circle times its area: (θ / 360) × πr². For r = 12 and θ = 60°, that's one-sixth of π × 144, about 75.40.

What is the difference between a sector and a segment?

A sector is the full wedge, bounded by two radii and the arc, and it includes the centre of the circle. A segment is only the region between the chord and the arc — what's left of the sector once you remove the triangle made by the two radii. The segment is always the smaller of the two.

How do you find the chord length of an arc?

The chord is the straight line across the ends of the arc. Its length is c = 2r sin(θ / 2), using half the central angle. When θ is 60°, the chord works out exactly equal to the radius.

What's the difference between arc length and chord?

The arc length is the curved distance along the edge of the circle; the chord is the straight-line shortcut between the same two endpoints. The arc is always longer than the chord, and the gap between them grows as the angle widens.

How do you find the radius from the arc length and angle?

Rearrange L = r θ to r = L ÷ θ, with the angle in radians. So a 30° arc (0.5236 rad) measuring 10 units long comes from a radius of about 19.1. This calculator does the degree-to-radian conversion automatically.