The Sector Area Calculator computes the area of a circular sector from its radius and central angle. It supports both degrees and radians and provides four modes: calculate sector area, find arc length, determine the angle from a known area and radius, or find the radius from a known area and angle.
Each calculation includes step-by-step working and an SVG diagram that highlights the sector within a circle. The calculator also shows the sector perimeter (arc length plus two radii) for each computation.
Your calculations will appear here
A sector is the region of a circle enclosed by two radii and an arc. The area of a sector is a fraction of the full circle area, proportional to the central angle.
In degrees, the sector area formula is A = (theta / 360) times pi times r squared. In radians, the equivalent formula is A = (1/2) times r squared times theta, which is often simpler to work with.
The arc length of a sector is the portion of the circumference subtended by the central angle. The formula is s = r times theta (where theta is in radians). In degrees, this becomes s = (theta / 360) times 2 pi r.
The perimeter of a sector is the arc length plus two radii: P = s + 2r. This is useful for calculating the boundary length of sector-shaped regions such as pizza slices or pie charts.
Sectors appear frequently in engineering (cam profiles, gear teeth), architecture (arched windows), and data visualisation (pie charts). The radian-based formula is preferred in calculus and physics because it simplifies differentiation and integration.
Problem: Find the area of a sector with radius 10 and central angle 60 degrees.
Solution: A = (60/360) times pi times 10 squared = (1/6) times pi times 100 = 100pi/6.
Answer: 52.3599
Problem: Find the arc length for a sector with radius 8 and angle pi/3 radians.
Solution: s = r times theta = 8 times pi/3.
Answer: 8.3776
Problem: A sector has area 25 and radius 5. What is the central angle?
Solution: theta = 2A / r squared = 2 times 25 / 25 = 2 radians.
Answer: 2 radians (114.592 degrees)
Problem: A sector has area 50 and central angle 90 degrees.
Solution: theta in radians = pi/2. r = sqrt(2A / theta) = sqrt(2 times 50 / (pi/2)) = sqrt(200 / pi * 2).
Answer: 7.9789
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