Arc Length Calculator: Formula, Sector Area & Chord

The arc length of a circle is the radius times the central angle in radians: L = r × θ. It is the distance along the curved edge of a slice of the circle. A 90 degree arc on a circle of radius 10 measures about 15.71 units. Enter the angle and the radius above, and the calculator finds the arc length, sector area, chord, and diameter at once.

It accepts the angle in degrees, radians, gradians, or turns, and lets you work backward from the arc, the chord, or the sector area too. Work in any unit from millimeters to yards, and copy the result in one tap.

What is arc length?

Arc length is the distance measured along the curved part of a circle, not straight across it. Picture a slice of pizza: the arc is the curved crust, while the two straight edges are the radii. The longer the slice angle, the longer the crust. Arc length depends on two things, how big the circle is and how wide the angle at the center is.

For a full circle, the arc length is the entire circumference. For any smaller angle, it is a matching fraction of that circumference. θ L c r

How do you find arc length?

The clean formula uses radians: L = r × θ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, either convert it to radians first, or use the degree version directly: L = 2 × π × r × (θ / 360).

For a radius of 10 and an angle of 90 degrees, the degree formula gives 2 × π × 10 × (90 / 360) = 62.83 × 0.25 = 15.71. The radian formula gives the same: 10 × (π / 2) = 15.71. The calculator handles the conversion, so you can enter degrees and it does the rest.

What is the arc length formula?

There are two forms, one for each angle unit:

• Radians: L = r × θ
• Degrees: L = 2 × π × r × (θ / 360)

They give identical answers. The radian version is shorter because radians are defined so that one radian of angle sweeps exactly one radius of arc. That is the whole reason radians exist, to make this formula as simple as r times θ. The degree version has the extra 360 because it converts the degree fraction into a slice of the full circumference.

How do you find arc length from degrees?

Take the angle as a fraction of a full circle, then multiply by the circumference. The fraction is the angle divided by 360. Multiply that by 2 × π × r. So a 45 degree arc on a radius of 8 is (45 / 360) × 2 × π × 8 = 0.125 × 50.27 = 6.28. In other words, a 45 degree arc is one eighth of the way around, so it is one eighth of the circumference. You can find that full circumference with the circle perimeter calculator.

Worked example: arc, chord, and sector together

Take a 120 degree slice of a circle with radius 5, like a wide wedge. First convert the angle to radians: 120 degrees is 120 divided by 180 times pi, which is 2pi over 3, about 2.094. The arc length is 5 times 2.094, about 10.47. The sector area is half times 25 times 2.094, about 26.18 square units. The chord is 2 times 5 times the sine of 60 degrees, which is 10 times 0.866, about 8.66. So the curved crust is 10.47, but the straight shortcut across is only 8.66. The calculator produces all three the instant you enter the angle and radius.

Arc length for common angles

Quick reference for a radius of 10. Scale proportionally for other radii, since arc length grows in direct proportion to the radius, or use the calculator for any values.

What is the sector area?

A sector is the full pie slice, the two radii plus the arc between them. Its area is A = ½ × r² × θ in radians, or the fraction θ / 360 of the whole circle's area in degrees. For a radius of 10 and a 90 degree angle, the sector area is a quarter of the circle, about 78.54 square units. The calculator shows the sector area alongside the arc length, since they share the same angle and radius.

What is the chord of an arc?

The chord is the straight line connecting the two ends of the arc, the shortcut across instead of around. It is always shorter than the arc. The chord length is c = 2 × r × sin(θ / 2). For a 90 degree arc on a radius of 10, the chord is about 14.14, noticeably less than the 15.71 arc. At exactly 180 degrees, the chord equals the diameter, since the arc is a half circle.

Why do radians make the formula simpler?

Radians are an angle unit built around the circle itself. One radian is the angle that sweeps an arc exactly equal to the radius. That definition is why arc length is just r times theta with no extra constants. A full circle is 2pi radians, because the circumference is 2pi times the radius, so it takes 2pi radius-lengths of arc to go all the way around. Degrees, by contrast, are an arbitrary split into 360 parts, which is why the degree formula needs that division by 360 to work. Radians remove the clutter.

Real example: distance around a running track curve

Running tracks show arc length in action. The curved ends of a standard track are semicircles, half-circle arcs of 180 degrees. If the inner lane has a radius of about 36.5 meters, each curved end has an arc length of pi times 36.5, roughly 114.7 meters. The two curves together add about 229 meters, and the two straights make up the rest of the 400 meter lap. Outer lanes have a larger radius, so their curves are longer, which is exactly why staggered starting lines exist. Runners in outer lanes start ahead to even out the extra arc length.

What units does arc length use?

Plain length units, the same as the radius. A radius in centimeters gives an arc length in centimeters. The angle is separate and can be degrees, radians, gradians, or turns. The sector area comes out in square units, since it is an area. Keep the radius and arc in matching length units, or let the calculator convert.

How does arc length connect to the full circle?

Arc length is simply a fraction of the perimeter of a circle. The whole way around is the circumference, 2 times pi times the radius. An arc is the portion of that edge cut off by the central angle. So a 90 degree arc is a quarter of the circumference, a 180 degree arc is half, and a 360 degree arc is the entire circumference. This makes the circle's perimeter the natural starting point: find the full circumference, then take the slice you need.

Where is arc length used in real life?

Anywhere something curves. Road and railway engineers use it to lay out curved sections and calculate how much track or asphalt a bend needs. It sizes curved architectural features, arches, and staircases. Clock makers use it for the sweep of a hand. It measures the distance a point travels on a rotating wheel or gear, and the length of a curved conveyor belt. Any curved path along a circle is an arc length problem.

Frequently asked questions

What is arc length?

It is the distance along the curved edge of a circle slice, found with L = r × θ when the angle is in radians. A 90 degree arc on a radius of 10 is about 15.71.

What is the formula for arc length?

L = r × θ in radians, or L = 2 × π × r × (θ / 360) in degrees. Both give the same result.

How do you find arc length without radians?

Use the degree formula: multiply the circumference by the angle over 360. This skips the radian conversion entirely.

What is the difference between arc length and chord?

Arc length is the curved distance around the edge. The chord is the straight line across between the two ends. The chord is always shorter.

Is arc length the same as circumference?

Only for a full circle. Arc length is a portion of the circumference matching the central angle. A 360 degree arc is the whole circumference.

How does arc length relate to radius?

It is directly proportional. Double the radius and the arc length doubles for the same angle, because a bigger circle has a longer edge to sweep.

How do you find the radius from arc length?

Divide the arc length by the angle in radians: r equals L divided by theta. The calculator does this when you enter the arc length and the angle.

What is a sector versus a segment?

A sector is the pie slice bounded by two radii and the arc. A segment is the smaller region between the chord and the arc, cutting off the triangle part. This calculator finds the sector area.

Can arc length be longer than the diameter?

Yes, easily. Even a half circle arc, at 180 degrees, is pi times the radius, which is longer than the diameter of twice the radius, since pi is bigger than 2. Larger angles give even longer arcs.

What is arc length in radians of 1?

It equals the radius. An angle of exactly one radian sweeps an arc equal to the radius, which is the definition of a radian. So for a radius of 10, a 1 radian arc is 10.

How many radians are in a full circle?

Two pi, about 6.283. This matches the circumference formula, since going all the way around sweeps 2pi radius-lengths of arc.

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