A circular sector's area is A = ½θr² when the central angle θ is in radians, and its arc length is s = rθ using the same θ and radius r. Both relationships come directly from the fraction of a full 2π rotation that the central angle covers, which is why the same θ drives both formulas. To apply them in practice, you need a positive radius, a central angle greater than zero but no greater than one full circle, and a clear choice between degrees and radians — because degrees must first be converted with θ_rad = θ_deg × π/180 before either formula is valid. The Sector Area Calculator handles that conversion automatically and also reports the percentage of the full circle the sweep represents, so a single set of inputs gives you the sector area, the arc length, and the share of the circle all at once. Everything runs locally in the browser, so there is nothing to upload and no account to create.
Defining a Circular Sector and Its Share of the Circle
A sector is the region bounded by two radii and the arc between them, often compared to a slice of pie. Unlike the full circle, a sector covers only the fraction of the rotation swept by its central angle. The OpenStax Algebra and Trigonometry chapter on angles establishes that every angle maps to a unique arc length on a given circle, which is the geometric reason a sector can be fully described by just two numbers: the radius r and the central angle θ. The sector's share of the full circle is therefore θ/(2π) when θ is in radians, and that same fraction drives both the area and the arc length. Understanding this share early helps you sanity-check results: a 90° sweep should always cover one quarter of the circle, no matter what units the radius uses, because 90° is exactly π/2 radians and π/2 divided by 2π is 1/4.
The Two Formulas Behind Sector Area and Arc Length
The sector area formula is A = ½θr², and the arc length formula is s = rθ, both with θ measured in radians. Because both depend on the same θ, the percentage of the circle and the geometry of the slice are linked in a single proportion. If your angle is given in degrees, convert first using θ_rad = θ_deg × π/180, then plug into either formula. A common student trap is to use the degrees value directly, which yields an answer that is too large by a factor of about 57.3 — the calculator performs the conversion for you, but if you are working by hand, the conversion step is the one you cannot skip. The same logic that produces A = ½θr² also produces s = rθ: a full circle has area πr² and circumference 2πr, and a sector with central angle θ captures exactly θ/(2π) of both. The SIU Math 250 review sheet on the area of a sector walks through that derivation independently, which is why both resources arrive at the same pair of formulas from different starting points.
Calculating Sector Area and Arc Length Step by Step
- Open the Sector Area Calculator and enter the positive circle radius using one consistent length unit — for example, 20 if your radius is in feet, or 6.1 if your radius is in meters.
- Enter the central angle and choose whether it is expressed in degrees or radians using the selector next to the angle field.
- Select Calculate sector, then read the area, arc length, and percentage of the full circle directly from the result panel.
That is the full procedure. The page does the rest: it converts degrees to radians when needed, computes 0.5 × θ × r² for the area, computes θ × r for the arc length, and reports θ/(2π) as the share of the full circle. Changing an input clears the old result, so a stale area from a previous entry is never mistaken for the new result. If the calculator rejects an entry, the most likely cause is a non-positive radius, an angle of zero, or an angle larger than 360° (or 2π radians); correct the input and recalculate rather than guessing at a workaround.
Units, Precision, and Input Limits
The calculator keeps whatever length unit you entered — if you put the radius in centimeters, the area comes out in square centimeters and the arc length in centimeters; if you put it in feet, both come out in feet. The tool does not perform unit conversion, so do not mix a radius in one unit with an expected result in another. The displayed numbers use up to 12 significant digits, which is enough to balance precision with readability; any decimal that contains π is an approximation because JavaScript evaluates π numerically rather than symbolically. Input validation rejects zero, negative, non-finite, or extremely large radii, and the central angle must be greater than zero and no greater than one full circle (360° or 2π radians). Everything runs locally in the browser with no uploads and no account requirement, and the formula statement stays visible beside the result so the calculation boundary is easy to audit at a glance.
Sector vs Segment and Other Related Shapes
A sector is not the same as a segment, and the difference matters when you choose a tool. A sector is bounded by two radii and the arc between them — the pie-slice shape. A segment is bounded by a chord (a straight line) and the arc, so its area is not a simple fraction of πr² and needs the chord length or sagitta to compute. The Sector Area Calculator returns the ordinary sector swept by a positive central angle; it does not return a segment, an annulus sector with inner and outer radii, an elliptical sector, a reflex sweep beyond one rotation, or an area from chord and sagitta measurements.
| Shape | Bounded by | Use this calculator? |
|---|---|---|
| Circular sector | Two radii + arc | Yes |
| Circular segment | Chord + arc | No |
| Annulus sector | Two inner radii + two outer radii + two arcs | No |
| Elliptical sector | Two radii of an ellipse + arc | No |
| Reflex sweep (> 360°) | Two radii + arc beyond one rotation | No |
If your geometry calls for a full circle instead of a slice, the Circle Area Calculator handles A = πr² and the Circumference Calculator handles the full perimeter. The Annulus Area Calculator is the right tool when the shape is a ring with an inner and outer radius rather than a single-radius slice. None of those overlap with the Sector Area Calculator, so pick the tool whose known measurements match your shape rather than forcing one formula onto another geometry.
Worked Example: A 30-Degree Sprinkler Sweep at 20 ft
OpenStax gives a worked example of a sprinkler with radius 20 ft sweeping 30°, and the same numbers double as a useful check for the calculator. Following the formula logic step by step:
- Convert the angle to radians: θ = 30 × π/180 = π/6.
- Sector area = 0.5 × (π/6) × 20² = 0.5 × (π/6) × 400 = 200π/6 = 100π/3 ≈ 104.7197551 square feet.
- Arc length = (π/6) × 20 = 20π/6 = 10π/3 ≈ 10.47197551 feet.
- Percentage of the full circle = (π/6) / (2π) = 1/12 ≈ 8.33333%.
Entering radius 20 and angle 30 in degree mode on the Sector Area Calculator reproduces these values and also gives the same arc length and percentage, because the page applies the same three formulas in the same order. If you want to double-check by hand, the SIU Math 250 review sheet independently derives A = ½r²θ and s = rθ from the fraction of the rotation, which is why the textbook, the review sheet, and the calculator all agree to the digit shown.
Common Central Angles at a Glance
A handful of angles come up often enough to be worth memorizing, and their radian equivalents are exact so they make a useful sanity check when you switch between degree and radian mode.
| Degrees | Radians | Share of full circle |
|---|---|---|
| 30° | π/6 | 1/12 ≈ 8.33% |
| 45° | π/4 | 1/8 = 12.5% |
| 60° | π/3 | 1/6 ≈ 16.67% |
| 90° | π/2 | 1/4 = 25% |
| 180° | π | 1/2 = 50% |
| 360° | 2π | 1 = 100% |
If your answer at 90° does not come out to one quarter of πr² in area and one quarter of 2πr in arc length, the most likely cause is forgetting the degree-to-radian conversion. Plugging θ = 90 directly into A = ½θr² gives 45r², which is roughly 57 times the correct value of πr²/4 ≈ 0.7854r², and the calculator avoids that trap by handling the unit choice explicitly through the angle selector.
When to Verify Against a Professional Method
For homework, sketching, and most everyday planning, the calculator is enough. For construction, surveying, machining, legal boundaries, or safety-critical engineering, retain appropriate significant figures and verify the result with the governing professional method and tolerances. Floating-point arithmetic in the browser is accurate to roughly 15 to 17 significant digits, but the displayed 12-digit result is only as reliable as the input you typed; a typo in the radius shows up cleanly because changing an input clears the old answer. If the problem involves a segment rather than a sector, or an annulus rather than a single ring, switch tools rather than forcing the sector formula onto a shape it was not designed for. The guide to circumference and area of a circle is a useful next read when your known measurements describe a full circle instead of a slice.