A numerical walkthrough
A 120° sector covers one third of a circle. At radius 7.5, full area is 56.25π; one third is 18.75π, approximately 58.9049 square units. This Sector Area example can be compared with length of the curved edge.
Math calculator
Sector Area Calculator
Calculate the slice of a circle enclosed by two radii and their arc. Formula and sector area remain visible in one place for independent verification.
Sector Area inputs
Calculation inputs
Practical meaning
Sector area models circular slices in mechanical rotation, pie-chart geometry, irrigation coverage, fan-shaped material, and many classroom circle problems.
Mathematical foundation
A circular sector occupies the same fraction of the disk that its central angle occupies of 360°. The area is therefore θ/360 times πr².
Reading sector area correctly starts with the mathematical structure described here.
Calculating it by hand
Square the radius and multiply by π to obtain full-circle area. Multiply that result by the central-angle fraction θ/360.
Before relying on the answer
Sector area includes the region from the center to the arc. A circular segment—the region between a chord and arc—requires subtracting a triangle and is a different calculation. If the Sector Area assumptions do not fit, consider complete circle area.
An impossible sector area sign or magnitude usually points to field assignment before it points to rounding.
Reading sensitivity and precision
The input labels—radius and central angle—encode the model used on this page. Write those labels beside source values when transferring a problem from paper or a spreadsheet. That small step catches transposed quantities and mixed units before they become a polished-looking sector area.
Sector area is linear in angle but quadratic in radius. Doubling the angle doubles area; doubling radius with angle fixed quadruples it. A one-field trial makes this relationship visible without reworking the entire example.
A sector reaches the center through two radii. A segment cuts off a cap with a chord, so segment area requires a triangle subtraction not present here. The distinction determines whether this page fits the original question.
Choose rounding after considering how the result will be used. Comparison may need only a few significant digits, while a later multi-step calculation benefits from carrying more. In either case, retain the page's formula with the sector area so the underlying definition remains visible.
If the number moves into a spreadsheet, give its cell a sector area heading and retain the source values radius and central angle nearby. Context matters more than extra displayed digits.
Answers to common follow-ups
Is a 90° sector one quarter of a circle?
Yes, so both its arc and area are one quarter of the corresponding full-circle measures.
What units does sector area use?
Square units derived from the radius unit.
Can the angle be 360°?
Yes. The result is the full circle area.
Is a sector the same as a segment?
No. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.