Sector Area Calculator

How to use the sector area calculator

What this calculator does

This calculator solves the linked measurements of one circular sector. Supply the Central angle and any one positive size measurement: Radius, Diameter, Sector area, Arc length, or Chord length. The most recently edited size field becomes the active input, and the other four are recalculated from the same canonical radius. The results are geometric identities subject only to decimal display rounding; they do not include material thickness, kerf, construction tolerance, or measurement uncertainty.

When to use it

Use it to check circle-sector exercises, convert a known arc or chord into a radius, size a fan-shaped piece of sheet material, compare cake or pizza slices, or prepare circular layouts. It is also useful when a drawing is dimensioned in degrees but a formula requires radians. OpenStax explains the relationship among radian measure, arc length, and sector area.

How to calculate

1. The calculator opens with a ready demonstration: Central angle 120°, Radius 6 cm, Diameter 12 cm, Sector area 37.6991118431 cm², Arc length 12.5663706144 cm, and Chord length 10.3923048454 cm. Radius is the active size input, and the example Excel workbook is available immediately.
2. Choose Central angle unit as Degrees (°) or Radians (rad), then enter the angle. Choose Measurement unit as millimeters, centimeters, meters, inches, or feet. Unit changes convert the active measurement and recalculate every dependent field.
3. Edit exactly the size measurement you know. For example, typing into Diameter makes Diameter the active input; typing into Arc length makes Arc length active. The blue-highlighted field identifies the current size driver. You do not need to clear the other measurements because the calculator overwrites them with consistent values.
4. Read the live Sector area result and supporting values, inspect the measurement table, and select Download Excel for a fresh validated OOXML workbook. Reset clears the demonstration, all six numeric fields, calculated outputs, table rows, and workbook state. Export remains disabled until a valid angle and one valid size measurement are entered again.

Input guide

Central angle is required. Enter a positive decimal no greater than 360 degrees or 2π radians. A realistic example is 90°. Increasing the angle increases sector area and arc length for a fixed radius. Chord length increases up to 180° and then decreases because the endpoints move back toward each other. Use a decimal point; an ambiguous decimal comma such as “1,5” and scientific notation such as “1e3” are rejected.

Central angle unit is required. Switching between Degrees (°) and Radians (rad) converts the current angle, so 180° becomes approximately 3.1415926536 rad. NIST identifies the radian as the SI derived unit for plane angle.

Measurement unit is required and applies to Radius, Diameter, Arc length, and Chord length; Sector area uses the corresponding square unit. Changing from centimeters to meters divides a length input by 100 and divides an area input by 10,000. A common mistake is changing the label mentally without converting the number; this control performs the conversion automatically.

Radius is an optional size input. Enter a positive length such as 6 cm. When active, doubling Radius doubles Diameter, Arc length, and Chord length but multiplies Sector area by four. Do not enter a diameter here.

Diameter is an optional size input. Enter a positive length such as 12 cm. The calculator divides it by two to recover the radius. Diameter is useful when a drawing specifies the full width of the circle.

Sector area is an optional size input. Enter a positive square-unit value such as 37.6991 cm². The calculator solves r = √(2A/θ). Be sure the selected Measurement unit matches the square unit of the area; selecting centimeters means the area entry is interpreted as cm².

Arc length is an optional size input. Enter a positive curved-boundary length such as 12.5664 cm. The calculator solves r = L/θ, with θ in radians. Do not substitute the straight chord for the curved arc.

Chord length is an optional size input. Enter the positive straight-line distance between the arc endpoints, such as 10.3923 cm. The calculator solves r = c/[2 sin(θ/2)]. At a full 360° turn, chord length is zero for every radius, so a chord cannot uniquely determine the circle; use Radius, Diameter, Sector area, or Arc length instead.

Output guide

The summary pills labeled Angle, Radius, Circle share, and Area provide a compact current-state check. Sector area measures the region bounded by the two radii and intercepted arc, in square units. Diameter is exactly 2r. Arc length is the curved boundary rθ. Chord length is the straight endpoint distance 2r sin(θ/2). The result labeled Central angle is the radian value actually used by the formulas. Circle share is θ divided by one full turn, shown as a percentage; 25% is a quadrant, 50% a semicircle, and 100% a full circle.

The measurement table's Metric, Value, Unit, and Formula columns repeat the canonical values used by the results and workbook. A zero chord at 360° is valid because the endpoints coincide. Negative outputs are never valid for this model. Very high values usually indicate a unit mismatch or an incorrectly entered size measure.

Worked example

For the startup demonstration, θ = 120° = 2.0943951024 rad and r = 6 cm. Sector area is ½ × 6² × 2.0943951024 = 37.6991 cm². Arc length is 6 × 2.0943951024 = 12.5664 cm. Chord length is 2 × 6 × sin(60°) = 10.3923 cm, and Diameter is 12 cm. Since 120° is one third of 360°, Circle share is 33.33%. Entering 12 in Diameter, 37.6991118431 in Sector area, 12.5663706144 in Arc length, or 10.3923048454 in Chord length reproduces the same circle within input precision. Wolfram MathWorld provides additional circular-sector identities and terminology.