Perimeter of a Sector Calculator
Find the total boundary of a circular sector from its central angle and radius, with automatic degree – radian and length-unit conversion.
Sector inputs
Enter a positive decimal, up to one full turn. Commas and scientific notation are not accepted.
Use a positive length. The selected unit applies to the radius and every length result.
Live results
28 cm
14 cm
14 cm
2 rad
50%
28 cm = 2 × 7 cm + 14 cm
Boundary breakdown
| Boundary component | Length | Share of perimeter |
|---|---|---|
| Two straight radii | 14 cm | 50% |
| Circular arc | 14 cm | 50% |
| Total perimeter | 28 cm | 100% |
The perimeter consists of exactly two straight radius edges plus the curved arc. Shares are calculated from the same unrounded values used for the main result and Excel workbook.
How to use the perimeter of a sector calculator
What this calculator does
This calculator finds the total distance around a circular sector: the curved arc plus the two straight radius edges that connect the arc to the circle's center. It also reports the arc length, the combined length of the two radii, the central angle in radians, and the arc's percentage of the complete boundary. It is a geometry calculator, not a drawing or surveying tool, so its accuracy depends on the angle and radius you provide.
When to use it
Use it when checking textbook geometry, estimating the edge length of a fan-shaped panel, planning trim around a sector-shaped sign or garden bed, or verifying a CAD or fabrication calculation. The formula applies to a minor sector, semicircle, major sector, or full circular turn as long as the central angle is greater than zero and no more than one revolution.
How to calculate
- The calculator opens with a ready-to-use demonstration: a central angle of 2 radians and a radius of 7 cm. Its example workbook is immediately available through Download Excel.
- Replace the Central angle with your positive decimal value. Choose Degrees or Radians; switching units converts the current value rather than changing its geometric meaning.
- Replace the Radius and choose its length unit. The same unit is used for the arc length, the two-radii contribution, and the perimeter.
- Read the live Perimeter and supporting results. Use the boundary table to compare the straight and curved portions.
- Select Download Excel to create a current-state workbook. Reset clears the demonstration and all calculated content; it may disable Excel export until both required values are complete and valid again.
Input guide
Central angle is required and accepts a plain positive decimal without commas or scientific notation. In degrees, the allowed range is more than 0° through 360°; in radians, it is more than 0 through 2π. A realistic example is 65°. A larger angle lengthens the arc and therefore raises the perimeter, while the two-radius contribution stays unchanged. A common mistake is entering a radian value while Degrees is selected.
Angle unit is required and determines how the central-angle number is interpreted. Degrees are familiar for construction and classroom diagrams; radians connect directly to arc length. The relationship is 180° = π radians. OpenStax's explanation of radian measure and circular arc length shows why the radian form makes the arc formula especially direct.
Radius is required and accepts a plain positive decimal. An example is 9 cm. Increasing the radius scales both the arc and the two straight edges, so the perimeter rises in direct proportion. Do not enter the diameter: if the measured width across the full circle is 18 cm, the radius input is 9 cm.
Length unit is required and supports millimeters, centimeters, meters, inches, and feet. Changing it converts the current radius and keeps the physical sector unchanged. Use one consistent unit for all measurements. NIST's overview of SI measurement units provides context for metric length units and prefixes.
Output guide
Perimeter is the exact geometric identity P = 2r + L, displayed in the selected length unit. Arc length is the curved portion L = rθ, where θ is in radians. Two radii is 2r, the combined straight-edge contribution. Angle in radians is the converted angle used in the formula. Arc share of perimeter is L ÷ P expressed as a percentage; it approaches 0% for a very small positive angle and rises as the angle grows. The displayed formula substitutes the current numbers, while the boundary table lists each component's length and percentage. These outputs are calculated identities, not statistical estimates.
Worked example
With the startup values θ = 2 radians and r = 7 cm, the arc length is L = rθ = 7 × 2 = 14 cm. The two straight radii total 2r = 14 cm. Therefore, P = 14 + 14 = 28 cm. The arc contributes 14 ÷ 28 = 50% of the boundary. The same values appear in the live cards, breakdown table, and startup Excel workbook.
Formula and interpretation
The sector boundary has three pieces: one circular arc and two radius-length segments. If the angle θ is already in radians, the arc length is L = rθ. Substituting this into P = 2r + L gives the compact formula below.
For an angle entered in degrees α, convert it first: θ = απ ÷ 180. That gives P = 2r + rαπ ÷ 180. A full 360° sector has an arc equal to the complete circumference, but its boundary definition here still includes both radius edges; for a physical full disk, those coincident internal edges are normally not counted. Use the calculator's full-turn result only when your problem explicitly defines the sector perimeter with the standard 2r + arc convention.
For another authoritative treatment of the relationship between rotation, radius, and arc distance, see OpenStax's angle of rotation and arc-length discussion. NIST also identifies the radian as the coherent SI unit for plane angle.
Common mistakes and useful checks
- Using degrees directly in L = rθ: that formula requires radians. The calculator performs the conversion automatically.
- Entering diameter as radius: this doubles every length result. Divide the full circle width by two before entering it.
- Mixing units: a radius measured in inches cannot be combined with an arc expected in centimeters without conversion. The single length-unit control keeps outputs consistent.
- Confusing sector perimeter with arc length: the perimeter includes the arc and both straight sides, so it is always greater than the arc alone for a positive radius.
- Sanity check: because P = r(2 + θ), doubling the radius while keeping the angle fixed must exactly double the perimeter.