How this instrument works
A circular sector's area answers a different question than its boundary does: how much surface sits inside the wedge, not how far the edge runs to enclose it. The formula A = ½r²θ, with θ measured in radians, echoes the proof Archimedes used to find a whole circle's area: he showed a circle equals a right triangle with one leg the radius and the other leg the full circumference, giving area ½ · r · 2πr = πr². A sector is that same triangle trimmed down to just its own slice of the rim — swap the full circumference for the sector's arc length rθ, and ½ · r · (rθ) collapses to ½r²θ.
The two inputs behave nothing alike inside that formula. Widen the angle from 1 radian to 2 and the area exactly doubles, because θ enters as a plain multiplier on a fixed circle. Widen the radius from 1 unit to 2 instead and the area quadruples, since r is squared — a genuinely bigger circle, not just a bigger slice of the same one. A sector's perimeter, rθ + 2r, scales the same way in both variables, so the area formula and the perimeter formula built from the same two numbers don't even agree on which input behaves 'normally.'
Push the central angle to its two extremes and the formula still behaves exactly as a sector should. At θ = 0 the wedge closes to a bare line with zero area — no vanishing residue, just 0 outright. At a full turn, θ = 2π radians (360°), the wedge is the entire disk and ½r²θ reduces to ½r²(2π) = πr², the ordinary circle-area formula recovered as this one's own upper limit rather than a separate rule to memorize.
- Enter the wedge's Radius — the straight-line distance from the center to the rim — in any length unit you like.
- Enter Central angle and pick its unit from the selector beside the field: degrees, radians, or turns are all accepted.
- Read Area for the surface enclosed by the wedge, reported in that length unit squared.
- To sanity-check a full circle, set Central angle to 360 degrees and compare Area against πr² worked out by hand.
Worked example — a 5-unit wedge at 1.2 radians
Picture a pie-slice cut from a circle of radius 5, its central angle fixed at 1.2 radians — about 68.75° for anyone converting in their head. Because θ already sits in radians, no conversion is needed before multiplying: A = ½ × 5² × 1.2 = ½ × 25 × 1.2 = 15 square units exactly, the figure this sheet returns for Radius 5 and Central angle 1.2 rad, matching the oracle used to verify every figure on this page.
That same wedge's boundary, arc plus two straight sides, works out to 16 on the sibling perimeter-of-a-sector calculator — close to the area's 15 purely by coincidence at this particular angle, since one quantity scales in squared units and the other in plain length and there is no reason for them to track each other.
Widen that radius-5 wedge to a full turn, θ = 2π ≈ 6.283185 radians, and the area climbs to ½ × 25 × 2π = 25π ≈ 78.539816 square units — precisely the ordinary circle-area result for radius 5, confirming the sector formula swallows the whole-circle case as its own boundary rather than needing a separate rule.
Questions
What is the formula for the area of a sector?
A = ½r²θ, where r is the radius and θ is the central angle in radians, not degrees. Square the radius, multiply by the angle, then halve the result. Enter degrees or turns into the Central angle field and the calculator converts to radians internally, so the field itself never forces that arithmetic on you by hand.
Why does the central angle have to be in radians?
Because ½r²θ is really πr² scaled by the fraction θ⁄2π of a full turn, and that fraction only equals θ⁄2π when θ is measured in the same radian units a full turn uses, 2π. Plug degrees straight into the formula and the hidden '360' throws the scaling off by a factor of about 57.3, since one radian is roughly 57.2958°.
How is sector area different from sector perimeter?
Area, ½r²θ, measures the surface inside the wedge in squared units. Perimeter, rθ + 2r, measures the boundary length in plain units on a separate calculator. For the radius-5, 1.2-radian wedge worked above, area comes to 15 while perimeter comes to 16 — close at that one angle purely by coincidence, not because the two quantities are related.
What happens to the area at a full 360° turn?
The wedge becomes the entire disk and ½r²θ reduces to ½r²(2π) = πr², the ordinary circle-area formula. For radius 5 that is 25π ≈ 78.5398 square units, the same figure a plain circle-area calculator would give for that radius — a full-turn sector isn't really a sector anymore, just the whole circle wearing the sector's formula.
What is the most common mistake people make with this formula?
Feeding in degrees where the formula wants radians. A central angle of 90° dropped straight into ½r²θ as if it were already radians overstates the area by a factor of roughly 57, since it skips the degree-to-radian conversion entirely. This calculator's Central angle field accepts degrees, radians, or turns and converts internally, but a hand calculation with this same formula needs that conversion done first.
Does the formula still work for a central angle larger than a full turn?
Arithmetically θ can exceed 2π and ½r²θ keeps growing, but geometrically that no longer describes a simple wedge — past 360° a 'sector' would have to overlap ground it already swept. Treat any angle beyond one full turn, 2π radians or 360°, as outside this formula's intended geometric meaning even though the arithmetic itself raises no objection.