Circle Sector Calculator

What a circle sector is

A sector is the slice of a circle bounded by two radii and the arc between them. It is the circular version of a pie slice, which is why searches often mention sector area, sector formula, and circle sector calculator together.

This page focuses on the full sector result: area, arc length, chord length, perimeter, and the angle in both degrees and radians. It is useful when the geometry problem gives you the radius and central angle and you want the exact slice measurements without rearranging the formulas yourself.

That fuller result matters because many sector questions start with area but quickly turn into layout checks such as how wide the straight span is, whether the slice is a minor sector or major sector, and how much boundary length needs to be measured or cut.

How the formulas fit together

The sector formulas all use the same central angle. When theta is in radians, the area is one-half times r squared times theta, the arc length is radius times theta, and the perimeter is the arc plus the two radii that bound the sector.

Chord length comes from the same geometry: chord = 2r sin(theta / 2). That gives the straight-line distance across the sector opening, which is often the missing measurement when a worksheet or fabrication sketch shows the curved edge and the end points together.

That is why the calculator shows all the values together. A sector area search often leads to a follow-up arc length or chord length question, and the perimeter is the final check when you need the whole outline of the slice.

Degrees versus radians

Radians are the natural unit for sector formulas because the area and arc formulas use theta directly. If your angle is already in radians, you can plug it straight into the formulas.

If your angle is in degrees, convert it first with theta × π / 180. For example, 90 degrees is π/2 radians, 180 degrees is π radians, and 60 degrees is π/3 radians. The calculator handles that conversion automatically.

Common sector checkpoints

A few central angles come up repeatedly in homework, layout, and fabrication checks. A 30-degree sector is one-twelfth of the circle, a 45-degree sector is one-eighth, a 60-degree sector is one-sixth, and a 90-degree sector is a quarter circle. Seeing those checkpoints early makes it easier to sanity-check whether the sector area, arc length, and chord length feel proportional before you rely on the final answer.

A 180-degree sector is a semicircle, while a 240-degree sector is a major sector that covers two-thirds of the circle. Those two examples are useful because they show how the same radius can produce either a straight-diameter checkpoint or a large wraparound slice depending on the central angle.

That is also why a circle sector calculator benefits from quick angle presets rather than only raw input boxes. Common geometry tasks often start with a named shape such as a quadrant, semicircle, or 60-degree wedge, and the preset angle makes the rest of the workflow faster to verify.

Chord length and sector type

Chord length is the straight-line distance between the endpoints of the arc. It is smaller than the arc length for every ordinary sector, but it becomes especially useful when you are checking a layout, a drawing, or a shape comparison against a circular segment.

Sector type depends on the central angle. Angles less than 180 degrees produce a minor sector, 180 degrees produces a semicircle, and angles greater than 180 degrees but less than 360 degrees produce a major sector. Seeing the sector type beside the numbers helps you sanity-check whether the result matches the shape shown in the diagram.

Worked examples

For a radius of 10 and a 90-degree angle, the sector area is 78.5398 square units, the arc length is 15.708 units, the chord length is 14.1421 units, and the perimeter is 35.708 units. The sector is one-quarter of the circle, so it also represents 25 percent of the full area.

For a radius of 8 and a 60-degree angle, the sector area is 33.5103 square units, the arc length is 8.3776 units, the chord length is 8 units, and the perimeter is 24.3776 units. Using radians instead of degrees gives the same result once the angle is converted correctly.

For a radius of 9 and a 240-degree angle, the sector area is 169.646 square units, the arc length is 37.6991 units, the chord length is 15.5885 units, and the perimeter is 55.6991 units. Because the angle is greater than 180 degrees, the calculator classifies it as a major sector.

Sector area versus segment area

A sector is bounded by two radii and an arc. A segment is the region between a chord and an arc, which is a different shape and needs a different formula. If you only need the straight-line distance across the arc, use the arc length calculator or chord-related tools instead.

This distinction matters because a sector calculator answers the slice question, while a segment calculator answers the curved cap question. The names sound similar, but the geometry is not the same.

Showing chord length on a sector page helps with this distinction. The chord belongs to both sector and segment problems, but the area formula does not. A segment area problem subtracts the triangle under the chord, while a sector area problem keeps the entire slice between the two radii.

When to use this calculator

Use this page when you need the full sector result from a radius and angle: area, arc length, and perimeter together. If you only need area, the sector area calculator is a narrower fit. If you only need arc length, the arc length calculator is the better match.

The circle sector calculator also helps when a problem is written in degrees but the formula wants radians, because it keeps both units visible so you can check the conversion after the result is calculated.

It is also the better fit when you want a quick worksheet-style answer with the chord length and sector type shown at the same time. That makes it easier to verify whether the slice is a minor sector, a semicircle, or a major sector before you continue.

Limitations

This calculator is intended for true sectors with an angle greater than 0 and less than 360 degrees. A full circle is a different problem, and the sector perimeter becomes ambiguous at that point.

The calculator assumes a perfect Euclidean circle. It does not estimate irregular arcs, measured tolerances, or fabrication allowances, and it does not try to unwrap multiple revolutions into one result.

It also assumes the radius and central angle describe the same sector directly. If your problem gives arc length, diameter, or a segment height instead, a different calculator may be the more direct route.