BYTETOOLS

Circle Sector and Segment Calculator

Solve a circular sector or segment from any two known values: arc length, chord, sector area, segment area, segment height and the major sector are all returned.

5
Radius
60
Angle (degrees)
5.236
Arc length
13.09
Sector area

Sector and segment

  • Central angle = 1.047198 rad = 60°
  • Arc length s = rθ = 5.235988
  • Chord = 2r·sin(θ/2) = 5
  • Sector area = ½r²θ = 13.089969
  • Segment area = ½r²(θ - sin θ) = 2.264652
  • Segment height = r(1 - cos(θ/2)) = 0.669873
  • Sector perimeter (two radii + arc) = 15.235988
  • Segment perimeter (arc + chord) = 10.235988

The rest of the circle

  • Major central angle = 300°
  • Major arc length = 26.179939
  • Major sector area = 65.449847
  • Full circle circumference = 31.415927
  • Full circle area = 78.539816
  • This sector is 16.667% of the circle

What is the Circle Sector and Segment Calculator?

The ByteTools Circle Sector and Segment Calculator solves the whole slice from whichever two measurements you have.

  • Five solving modes, including chord plus segment height for arch and pipe work
  • Arc length, chord, sector area, segment area and segment height in one pass
  • Major sector and major arc shown alongside the minor slice
  • Degrees or radians, with the angle reported in both
  • Rejects impossible input such as an arc longer than the circumference
  • 100% private — every value stays in your browser

How to use the Circle Sector and Segment Calculator

  1. 1

    Choose which two measurements you know from the first dropdown.

  2. 2

    Set the angle unit to degrees or radians.

  3. 3

    Enter your two values in the fields that appear.

  4. 4

    Read the radius, angle, arc length and sector area in the summary tiles.

  5. 5

    Open the two result panels for the segment figures and the rest of the circle, then copy the results.

About the Circle Sector and Segment Calculator

The ByteTools Circle Sector and Segment Calculator solves the whole slice from whichever two measurements you have. Enter a radius and angle, a radius and arc length, a radius and sector area, an arc length and angle, or even a chord and its segment height, and the rest is derived.

Results cover arc length s = rθ, sector area ½r²θ, chord 2r·sin(θ/2), circular segment area ½r²(θ - sin θ), segment height r(1 - cos(θ/2)), and the perimeters of both the sector and the segment. The major sector on the other side of the chord is shown too, along with what fraction of the full circle your slice takes up.

All the trigonometry runs in your browser with JavaScript, so nothing is uploaded and the tool works offline. Angles can be entered in degrees or radians, and impossible combinations — an arc longer than the circumference, for instance — are explained rather than rounded away.

Frequently asked questions

What is the formula for the area of a sector?

In radians it is simply ½r²θ. A radius of 5 and a 60° angle (π/3 radians) gives ½ × 25 × 1.0472 ≈ 13.09 square units. In degrees the same thing is written (θ/360) × πr², which is why a sector's area is just its share of the whole circle.

What is the difference between a sector and a segment?

A sector is the pie slice bounded by two radii and the arc. A segment is the smaller region cut off by the chord alone, so it is the sector minus the triangle between the two radii. Its area is ½r²(θ - sin θ).

How do you find the radius from a chord and its height?

Use r = (h² + (c/2)²) / 2h, where c is the chord and h is the height of the arc above it, also called the sagitta. A 6-unit chord with a 1-unit rise sits on a circle of radius 5. This calculator has a dedicated mode for exactly this measurement.

How do I calculate arc length?

Multiply the radius by the angle in radians: s = rθ. If your angle is in degrees, convert with θ = degrees × π/180 first. A 60° arc on a radius of 5 is 5 × π/3 ≈ 5.236 units, and the tool converts between the two units for you.

What is the segment height used for?

Also called the sagitta or rise, it is the perpendicular distance from the middle of the chord to the arc. Carpenters, sheet-metal workers and pipe fitters use it to set out arches and curved cuts from a straight measurement, since it is far easier to measure than an angle.

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