Arc Length and Sector Area Calculator

1. Enter the radius of the circle. If you only know the diameter, halve it first.
2. Enter the central angle, the angle at the center between the two radii.
3. Pick the angle unit, degrees or radians. Switching converts the angle for you.
4. Read the arc length, then check the sector area, chord, segment area, and perimeter below it.

The result uses the same length unit as your radius. A radius in inches gives an arc in inches and an area in square inches.

An arc is a piece of the circle's edge. Its length depends on the radius and on how wide the central angle is. With the angle in radians, the formula is very short.

A full turn is 360 degrees, or 2 pi radians. So a degree angle is a share of 360, and the arc is the same share of the full circumference.

Both forms give the same answer. To move between units, multiply degrees by pi / 180 to get radians.

A sector is the wedge between two radii and the arc, like a slice of pie. Its area is the same share of the full circle area as the angle is of a full turn.

A handy link between the two results is A = s x r / 2. Once you know the arc, the sector area is one multiplication away.

The calculator also finds the segment, the region between the chord and the arc. It is the sector minus the triangle formed by the two radii.

Take a circle with radius 10 and a central angle of 90 degrees, a quarter of the circle.

1. Convert the angle: 90 x pi / 180 = pi / 2, about 1.5708 radians.
2. Arc length: s = 10 x 1.5708 = 15.708 units.
3. Sector area: A = 1/2 x 10^2 x 1.5708 = 78.540 square units.
4. Chord: c = 2 x 10 x sin(45 degrees) = 14.142 units.
5. Segment area: 78.540 minus the triangle area of 50 = 28.540 square units.

The degree formula agrees: 2 x pi x 10 x 90 / 360 = 15.708. The sector is 25 percent of the circle, and its perimeter is 10 + 10 + 15.708 = 35.708 units.

The arc follows the curve. The chord cuts straight across from one end of the arc to the other. The chord is always the shorter path.

For small angles the two are almost equal. At 10 degrees on a radius of 10, the arc is 1.7453 and the chord is 1.7431. At 180 degrees the gap is large: the arc is pi x r, but the chord is just the diameter, 2r.

Values for a circle of radius 1. For any other radius, multiply the arc and chord by r, and the areas by r squared.

Pizza and Pie Slices

A 14 inch pizza has a radius of 7. Cut into 8 slices, each slice is a 45 degree sector with about 19.24 square inches of pizza and 5.50 inches of crust.

Curved Paths and Tracks

Garden edging, curved walkways, and running track bends all follow arcs. The arc length tells you how much edging, paving, or distance you need.

Fans and Sprinklers

A lawn sprinkler that sweeps 120 degrees waters a sector. The sector area tells you how much ground it covers at a given throw radius.

Wheels and Pulleys

When a wheel turns through an angle, a point on its rim travels an arc. That arc is how far the wheel rolls, so s = r x theta links rotation to distance.

1. Degrees in the Radian Formula

Putting 90 straight into s = r x theta gives 900 for a radius of 10, not 15.708. Convert degrees to radians first, or use the degree form of the formula.

2. Diameter Instead of Radius

Using the diameter doubles the arc length and makes the sector area four times too big. Always halve the diameter before you start.

3. Calculator in the Wrong Mode

The chord and segment formulas use sine. If your calculator is in degree mode while you type a radian angle, the sine will be wrong.

4. Mixing Up Sector and Segment

A sector includes the triangle at the center. A segment does not. Check which region your problem asks for before you pick a formula.

The formulas are exact for a perfect circle. Results are only as accurate as the radius and angle you enter.

Arc length is s = r x theta, where r is the radius and theta is the central angle in radians. If the angle is in degrees, use s = 2 x pi x r x (angle / 360).

Sector area is one half times r squared times theta, with theta in radians. In degrees, it is pi x r squared x (angle / 360), which is the same share of the full circle area.

A radian is defined so that an arc of length r subtends an angle of 1 radian. That makes s = r x theta exact in radians. In degrees you must first convert, or you get an answer 57.3 times too big.

Arc length is the distance along the curve. Chord length is the straight line between the two ends of the arc. The chord is always shorter, except that both are zero for a zero angle.

A sector is the pizza slice shape bounded by two radii and the arc. A segment is only the part between the chord and the arc. Segment area equals sector area minus the triangle area.

A minor sector has a central angle under 180 degrees. A major sector has an angle over 180 degrees. At exactly 180 degrees the sector is a semicircle, half of the circle.

Not for a single sector. A full circle is 360 degrees or 2 pi radians, so this calculator accepts angles above zero and up to that limit. Larger angles only wrap around the circle again.

• Wolfram MathWorld: Circular Sector (sector area and arc length).
• Wolfram MathWorld: Arc Length (definition of arc length).
• Wolfram MathWorld: Circular Segment (chord and segment area).
• OpenStax, Algebra and Trigonometry 2e, section 7.1 Angles (radians, arc length, and sector area).