Math · Geometry
Area of a sector calculator.
Sector area and circle segment area from a radius and an angle or an arc length.
Inputs
Sector and segment
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How do you find the area of a sector?
A sector is a slice of a circle between two radii. Its area is the same fraction of the whole circle as its angle is of a full turn: A = ½ r² θ with θ in radians, or A = (angle / 360°) × πr² in degrees. Knowing the arc length s instead, A = ½ r s. A circle segment is the part between a chord and its arc: A = ½ r² (θ − sin θ), which is the sector minus the triangle formed by the two radii and the chord.
Worked example. Radius 12 and a 75° angle: sector area 94.2478, triangle part 69.5467, segment area 24.7011, arc length 15.708. A 270° sector on radius 9 has an area of 190.8518, three quarters of the circle’s 254.469.
Frequently asked questions
What is the formula for the area of a sector?
A = ½ r² θ with the angle in radians, or A = (angle ÷ 360) × π r² with the angle in degrees.
How do I find sector area from the arc length?
Use A = ½ r s, half the radius times the arc length. A radius of 12 and a 15.708 arc give 94.2478.
What is the area of a circle segment?
The segment is the sector minus the triangle: ½ r² (θ − sin θ). For radius 12 and 75° that is 24.7011.
Does it work for angles over 180 degrees?
Yes, up to 360°. For a reflex angle the segment is the larger piece and the triangle part counts as negative, which the formula handles.
Sources
Checked
Sector and segment area formulas, arc length, chord and sagitta are from these MathWorld pages. Radian measure is as defined on the Radian page. Runs entirely in your browser. Nothing you type is sent or stored.
Scope
Results are exact mathematics rounded for display, so they are only as good as the measurements you enter. Any length unit works as long as every value uses the same one; areas come out in that unit squared and volumes in that unit cubed. For homework, DIY layout and estimating, not a substitute for a surveyor, engineer or your teacher's required method.