Arc Length Calculator

An arc length calculator turns a radius and a central angle into the curved distance along a circle. The formula is s = rθ with the angle in radians. A circle with a radius of 12 inches and a 75° central angle has an arc of 15.708 inches, a chord of 14.6103 inches and a sector area of 94.2478 square inches - just under 21% of the whole circle.

The Circle Calculator is the sibling page for whole-circle work: use it when you want the radius, diameter, circumference and area of the complete circle from any single value. Use this page instead whenever only part of the circle matters - a slice, a curve, a bend, a pie wedge - because arc length, chord, sector area and segment area all depend on the central angle as well as the radius.

How arc length works

Every arc is a fraction of the circumference of its circle. That fraction is set by the central angle: an angle of 90° covers a quarter of the circle, so its arc is a quarter of the circumference. Written with degrees, the relationship is:

Written with radians it collapses to something far simpler, because a radian is defined as exactly the angle whose arc equals the radius:

Both forms produce identical numbers. The degree version is easier to picture, the radian version is easier to rearrange, and the calculator accepts either through the degrees and radians toggle. Three companion formulas complete the picture of the same slice:

The chord is the straight shortcut between the two ends of the arc, the sector is the whole pie slice bounded by two radii and the arc, and the segment is the thin sliver the chord cuts off the outside of that slice.

Worked example: a 12-inch radius at 75 degrees

Suppose you are laying out a curved shelf bracket on a circle of radius 12 inches, sweeping through a central angle of 75°. Every number below comes straight from the formulas above.

1. Convert the angle. 75 × π ÷ 180 = 1.308997 radians.
2. Arc length. s = 12 × 1.308997 = 15.708 inches. The degree form checks it: 2 × π × 12 × (75 ÷ 360) = 15.708 inches.
3. Chord. c = 2 × 12 × sin(0.654498) = 14.6103 inches, about 1.098 inches shorter than the curve.
4. Sector area. A = 12² × 1.308997 ÷ 2 = 94.2478 square inches.
5. Segment area. A = 12² × (1.308997 − 0.965926) ÷ 2 = 24.7011 square inches.
6. Sagitta (rise). h = 12 − 12 × cos(0.654498) = 2.4798 inches.
7. Sector perimeter. P = 15.708 + 2 × 12 = 39.708 inches.

For context, the full circle has a circumference of 75.3982 inches and an area of 452.3893 square inches, and 75° is 20.833% of a full turn. Sure enough, 15.708 ÷ 75.3982 and 94.2478 ÷ 452.3893 both come out at 0.20833.

Arc, chord and sector area on a circle of radius 10

The table below fixes the radius at 10 units and varies the central angle, so you can see how quickly each quantity grows. Arc length and sector area rise in a straight line with the angle; the chord does not, because it peaks at 180° and then shrinks back to zero as the arc closes on itself.

Two details are worth noticing. At 60° the chord is exactly 10.000, the same as the radius, because a 60° sector contains an equilateral triangle. And at 270° the chord has fallen back to 14.142, the identical value it had at 90°, since both angles subtend the same pair of endpoints from opposite sides.

Solving backwards for the radius or the angle

The same relationship rearranges two ways, and the calculator has a mode for each. To find the radius from a known arc and angle, use r = s ÷ θ. An arc of 15 units sweeping through 40° converts to 40 × π ÷ 180 = 0.698132 radians, so r = 15 ÷ 0.698132 = 21.4859 units. This is the everyday case when you have measured a curve on the ground and know the turn it makes but cannot reach the center point.

To find the central angle from a known arc and radius, use θ = s ÷ r. An arc of 15 units on a circle of radius 10 gives θ = 1.5 radians, which is 1.5 × 180 ÷ π = 85.9437°. Because the answer arrives in radians first, this direction shows the definition of the radian very plainly: 1.5 radius lengths of curve means an angle of exactly 1.5 radians.

Arc length by radius and central angle

This table gives the arc length s = rθ directly for common radii and angles. It is unit-free: read the radius column in inches and the answers are inches, read it in feet and the answers are feet. For a radius not shown, scale proportionally - arc length is exactly linear in the radius, so a radius of 25 at 90° is 2.5 times the value in the 10 row, or 39.27.

Degrees, radians and the share of the circle

Most people arrive with an angle in degrees, but the underlying mathematics runs on radians. One degree is π ÷ 180 = 0.017453 radians, and one radian is 180 ÷ π = 57.29578°. The table below lists the conversions worth memorizing, along with the fraction of the circle each angle represents. That fraction is the number the degree form of the arc length formula actually multiplies the circumference by.

How to use this calculator

Start by choosing what you are solving for. In Arc length mode you enter the radius and the central angle and the tool returns the arc immediately. In Radius mode you enter the arc and the angle. In Angle mode you enter the arc and the radius. Switch between degrees and radians with the toggle beside the angle field; the value converts as you switch, so a 90 entered in degrees becomes 1.570796 in radians rather than being reinterpreted.

The unit buttons underneath are cosmetic labels only. They do not convert anything, they simply write "in", "ft", "cm" or "m" next to the answers so the output reads the way your drawing does. Whatever you choose, keep every length you type in that same unit. The preset angle buttons (30°, 45°, 60°, 90°, 120°, 180°) fill the field in one tap for the angles that come up most often in layout work and homework.

Below the headline number the calculator shows the chord, both areas, the sagitta, the sector perimeter and the share of the full circle, then a step-by-step trace of the arithmetic and a table repeating the calculation at nine standard angles for your radius. The sector diagram redraws live, which is the fastest way to catch an angle that was typed in the wrong unit: a "1.5" entered as degrees looks like a hairline sliver, while 1.5 radians fills more than a quarter of the circle.

Who this calculator is for

Students in geometry, trigonometry and precalculus meet arc length the moment radians are introduced, and this page shows both the degree and radian routes side by side so the conversion never becomes a black box. Woodworkers and metal fabricators need the arc to cut a curved trim piece to length, the chord to check a template against a straight edge, and the sagitta to verify a bend without ever locating the center of the circle. Landscapers and hardscapers use the sector perimeter to order edging for a curved bed and the sector area to order mulch or sod for it.

Machinists and CNC programmers convert between a swept angle and a travel distance constantly, since a rotary axis is specified in degrees while feed and tool paths are specified in linear units. Sewing and quilting patterns for circle skirts and fan blocks are pure sector geometry: the waist is one arc, the hem is a second arc on the same central angle with a larger radius. And anyone dividing a round object into equal parts - a pie, a pizza, a clock face, a color wheel - is really asking for arc length at 360° divided by the number of pieces.

Key terms

• Arc - a continuous piece of the circle itself. Its length is measured along the curve, not across it.
• Central angle - the angle at the center of the circle between the two radii that bound the arc. This is the θ in every formula on this page.
• Radian - the angle for which the arc length equals the radius. A full turn is 2π radians, so 1 radian is 57.29578°.
• Chord - the straight line joining the two endpoints of the arc. Always shorter than the arc, and equal to the diameter only when the angle is exactly 180°.
• Sector - the pie-slice region bounded by the two radii and the arc. Its area is r²θ/2.
• Segment - the smaller region between the chord and the arc, that is, the sector minus the triangle. Its area is r²(θ − sin θ)/2.
• Sagitta - the height of the arc above the middle of its chord, r − r cos(θ/2). Also called the rise, the versine or the middle ordinate.
• Minor and major arc - for any two points on a circle there are two arcs. The one under 180° is the minor arc, the one over 180° is the major arc, and together they add to the full circumference.

What changes the result

Only two inputs drive everything, but they behave very differently. Radius scales lengths in a straight line and areas as a square: doubling the radius doubles the arc and the chord but quadruples the sector and segment areas. That is why a modest measuring error on a large radius matters far more for a material order than for a cut length.

Central angle scales the arc and the sector area linearly, but the chord and the segment behave non-linearly because of the sine term. Between 15° and 90° on a radius of 10 the arc grows from 2.618 to 15.708, six times larger, while the chord grows from 2.611 to 14.142, only about 5.4 times larger. The gap widens steadily with the angle: the arc exceeds the chord by roughly 0.3% at 15° but by 11.1% at 90°.

The angle unit is the third and least obvious factor, and it is where most wrong answers come from. Typing 1.5 when you meant 1.5° but the toggle sits on radians overstates the arc by a factor of 57.3. Always glance at the unit label in the field and at the live diagram before trusting the number.

Practical tips

• Measure the diameter, not the radius. On a real object the widest distance across is easy to measure and the exact center is not. Halve the diameter and enter that.
• Use the sagitta when you cannot reach the center. Measure the chord and the rise on the workpiece, then adjust the radius in the calculator until the sagitta matches what you measured.
• Order the sector perimeter, not the arc, for edging. A curved bed needs the two straight radii covered as well as the curve, which is s + 2r.
• Add a waste allowance to curved cuts. The arc is the finished length; kerf, spring-back on a bend and trimming all consume extra material.
• Divide 360 by the count for equal pieces. Eight equal slices means 45° each, twelve means 30°, five means 72°.
• Sanity-check against the circumference. The arc can never exceed 2πr for a single turn. If it does, the angle unit is wrong.

Limitations and assumptions

This is exact geometry, but it describes one specific figure. Keep these boundaries in mind:

• It models a true circular arc with a single constant radius. Ellipses, parabolas, spirals and free-form curves do not follow s = rθ, and an ellipse has no closed-form arc length at all.
• It is a flat, two-dimensional figure. The path along a sphere, a cone or a helix needs three-dimensional geometry, not this formula.
• It is not the calculus arc length of a general function y = f(x), which is the integral of the square root of 1 + (dy/dx)².
• Results are rounded for display but calculated at full floating-point precision, so a manual recheck at 3.14 will differ slightly in the last digits.
• The tool does not convert units. It assumes everything you enter is already in one consistent unit.
• For an angle above 360° the arc becomes total swept travel rather than a geometric sector, so the chord, segment and sagitta stop being meaningful for that case.

How it compares to related calculators

This page answers "how long is this piece of the circle, and how much area does the slice cover?" Other tools on the site fit neighboring questions better:

• For the whole circle - radius, diameter, circumference or area from any single one of them - use the Circle Calculator.
• For the area of rectangles, triangles, trapezoids and other flat shapes, use the Area Calculator.
• For the sides and angles of the triangle hiding inside a sector, use the Triangle Calculator.
• For the right triangle formed by the radius, half the chord and the apothem, use the Pythagorean Theorem Calculator.
• To turn a share of the circle into a percentage, or the other way around, use the Percentage Calculator.
• To evaluate the sine and cosine terms in the chord and segment formulas by hand, use the Scientific Calculator.