A sector is the region of a circle bounded by two radii and the arc between them, and it is the shape of a slice of pie or a section of a fan. This sector area calculator takes the radius and the central angle and returns the sector area, the arc length and the area of the matching segment.
What Is the Sector Area Calculator?
A sector is the pie slice shape, and its area is simply a proportion of the whole circle. The proportion is the central angle divided by 360 degrees, so a quarter turn takes a quarter of the area and a right angle takes exactly one.
This calculator returns the sector area, the arc length along the curved edge, the area of the circular segment that sits between the chord and the arc, and the chord length itself.
How the Sector Area Formula Works
Sector area = ½ × r ² × θ, where θ is the central angle in radians
The formula follows from the area of a triangle, because a sector can be thought of as a very large number of thin triangles fanning out from the centre. Each has an area of one half r squared times its tiny angle, and adding them all up gives the sector.
The calculator also works in degrees, which is how angles are usually specified, and converts them to radians internally. The arc length comes from the same relationship with r in place of r squared.
Sector, Segment and Arc
A circular segment is the region between a chord and the arc it cuts off. The segment is smaller than the sector by the area of the triangle formed by the two radii and the chord, so subtracting that triangle gives the segment.
It is worth keeping the three terms separate because they are regularly confused. The sector is bounded by two radii and an arc, the segment by a chord and an arc, and the arc is only the curved line itself.
Practical Uses
Sectors describe cam profiles and the swept area of a rotating arm, which is how reach and coverage are calculated for fire monitors, irrigation heads, searchlights and cranes.
They also come up in land and engineering work: the area of a circular track’s turning section, pie-shaped plots in surveying, and the swept area used to work out how much material a cutting tool removes.
Frequently Asked Questions
How do I calculate the area of a sector?
Use one half times r squared times the angle in radians, or multiply the area of the full circle by the angle divided by 360. A 90 degree sector of a 5 metre radius circle is 19.63 square metres.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, so it is a pie slice. A segment is bounded by a chord and an arc, so it is the curved cap left after removing the triangle from the sector.
How do I find the arc length?
Multiply the radius by the angle in radians. Equivalently, take the circumference multiplied by the fraction of a full turn, which is the angle divided by 360.
What is the area of a 90 degree sector?
It is one quarter of the full circle, so a quarter of pi times r squared. With a radius of 5 metres that is 19.63 square metres.
Why is the major sector area the circle minus the minor sector?
A major sector and a minor sector together fill the whole circle exactly once, with no overlap. So subtracting the small piece from the full circle gives the large one, which is what the calculator does when you select the major sector.