This all-in-one online Sector Area Calculator allows you to calculate the missing parameters of a circular sector if any two parameters of the sector are known, except for the following parameter pairs: chord, arc and chord, area.
Circular Sector
A circular sector or circle sector, is the portion of a circle enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.
A sector with the central angle of 180° is often called a half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively.
In the diagram \(\theta\) is the central angle, \(r\) is the radius of the circle, \(s\) is the arc length of the sector, and \(c\) is the chord connecting the endpoints of the arc.
The sector arc length can be expressed using the following quite obvious formula:
$$s = r \cdot \theta,$$
where \(\theta\) is the angle in radians. The length of the chord is given by the formula:
$$c = 2r \cdot sin\frac {\theta}{2}.$$
Area of a Circle Sector
The total area of a circle is known to be \(\pi r^2\). The area of the sector can be obtained by multiplying the circle’s area by the ratio of the angle \(\theta\) (expressed in radians) and \(2\pi\) (because the area of the sector is directly proportional to its angle, and \(2\pi\) is the angle for the whole circle in radians). So we come to the following circular sector area formula:
$$A = \pi r^2 \cdot \frac {\theta} {2 \pi} = { \frac {r^2 \theta}{2}}.$$
The circle sector area in terms of the arc length \(s\) can be obtained by multiplying the total area of the circle \(\pi r^2\) by the ratio of \(s\) to the total circle perimeter \(2\pi r\):
$$A = \pi r^2 \cdot \frac {s} {2 \pi r} = { \frac {r s}{2}}.$$
These formulas are used in our Sector Area Calculator. With this calculator you can easily find all the parameters of a circle sector (\(r\), \(\theta\), \(c\), \(s\), \(A\)) if any two of these parameters are known, except for the following parameter pairs: chord, arc and chord, area. The reason for the latter is that knowledge of these pairs of parameters does not allow, in the general case, to unambiguously find the remaining parameters.
The angle \(\theta\) can be specified both in degrees and in radians. Please note that angles greater than 360 degrees (\(2\pi\)) are taken modulo 360 degrees (\(2\pi\)).
Note that we don’t specify units of measure in our calculator. We assume that area is measured in square units of length.
Related calculators
Check out our other geometry calculators such as Rhombus Area Calculator or Triangle Calculator.