Sector & Segment Calculator — Area, Arc Length, Chord

About the Sector & Segment Calculator — Area, Arc Length, Chord

A sector is a "pie slice" of a circle — the region bounded by two radii and the arc between them. A segment is the region between a chord and the arc it subtends. Together, these two shapes form the foundation of circular geometry and appear everywhere from pizza cutting to pie charts, clock faces to protractors.

The sector area formula is elegantly simple: A = ½r²θ, where r is the radius and θ is the central angle in radians. For degrees, it becomes A = (θ/360°)πr². The arc length is equally straightforward: L = rθ. The segment area — the crescent-shaped region between a chord and its arc — adds a subtraction: A_seg = ½r²(θ − sin θ).

The chord length connecting the two endpoints of an arc is c = 2r sin(θ/2), and the segment height (sagitta) is h = r − r cos(θ/2). These measurements are essential in engineering (calculating material for curved structures), surveying (measuring circular arcs), and data visualization (sizing pie chart slices).

This calculator handles both degrees and radians, computes all sector and segment properties from radius and central angle, and includes presets for common real-world scenarios: pizza slices (1/6 and 1/8 of a circle), clock positions (30° per hour), pie chart slices (25%, 33%), and semicircles. A reference table shows how common angles relate to fractions of the full circle.

When This Page Helps

Sector geometry shows up whenever a full circle is only partly used. Designers use it for pie charts and gauges, engineers use it for curved panels and rotating parts, and students meet it in every unit on radians and arc length. This calculator is useful because it keeps the entire family of related measurements together: the angle, arc, chord, sector, segment, and fraction of the whole circle.

That saves time and reduces conversion mistakes. If you work from degrees, you can still see the radian value the formulas actually use. If you are comparing a pizza slice, a clock sector, or a curved cutout in sheet material, the outputs make it easier to tell whether you care about the curved boundary, the straight chord, or the enclosed area before you commit to a measurement or drawing.

How to Use the Inputs

1. Select the length unit for the radius (mm, cm, in, m, or ft).
2. Choose the angle unit: degrees or radians.
3. Enter the circle's radius.
4. Enter the central angle (0° to 360°, or 0 to 2π radians).
5. Click a preset for instant results: pizza slice, clock, pie chart, or semicircle.
6. View sector area, arc length, chord length, segment area, perimeter, and fraction of the full circle.
7. Compare the sector area to the segment and remaining circle in the visual breakdown.
8. Check the reference table for common angle fractions and their real-world applications.

Example Calculation

Tips & Best Practices

• Sector area and arc length are both exactly proportional to the central angle — double the angle, double both values.
• A semicircle (180°) has a segment area equal to the area above the diameter, and a chord equal to the diameter itself.
• For small angles, the chord and arc length are nearly equal — the arc "flattens out" into a straight line.
• In radians, the formulas are cleaner: no π/180 conversions needed. Many scientific applications use radians by default.
• Pie chart design: each percentage corresponds to 3.6° (1% of 360°), making it easy to convert data to angles.

Sector Area Starts With Angle Share

A sector is simply a fraction of a circle determined by its central angle. That is why sector area and arc length scale directly with the angle: if the angle is one quarter of a full turn, the sector area and arc length are both one quarter of the full-circle values. This way of thinking is often faster than memorizing separate formulas because it connects each result to the same underlying idea of circular share.

Chord And Segment Tell A Different Story

The chord measures the straight-line shortcut between the arc endpoints, while the segment measures the curved cap trapped between that chord and the arc. Those are different geometric questions from the sector itself. In fabrication and construction, chord length matters when you need a straight brace or cut edge, while segment area matters when you need the material inside a curved profile. The calculator exposes both so you can see how the same radius and angle generate different useful dimensions.

Common Real-World Uses

The calculator is well suited to pizza-slice sizing, gauge sweep calculations, pie-chart design, and any curved panel or window detail based on a central angle. It is also a strong teaching aid for switching between degrees and radians because every output depends on the same angle input. The reference table makes it easy to compare familiar fractions like 30°, 45°, 90°, and 180° without reworking the formulas each time.

Frequently Asked Questions

• What is the difference between a sector and a segment?

  A sector is the "pie slice" bounded by two radii and an arc. A segment is the region between a chord and the arc above it. Sector area = ½r²θ. Segment area = ½r²(θ − sin θ). The segment is always smaller than its corresponding sector.

• How do I find the area of a pizza slice?

  Measure the radius (half the pizza diameter) and divide 360° by the number of slices. For an 18" pizza cut into 8 slices: r = 9", θ = 45°. Area = ½ × 81 × (π/4) ≈ 31.8 in².

• What is a sagitta?

  The sagitta (segment height) is the maximum distance from the chord to the arc. It equals h = r − r cos(θ/2). It's used in optics, architecture (arched windows), and road surveying.

• How do I convert between degrees and radians?

  Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees. Key values: 90° = π/2, 180° = π, 360° = 2π.

• Why is the segment area formula θ − sin θ?

  The segment area equals the sector area minus the triangle formed by the two radii and the chord. Sector = ½r²θ, triangle = ½r²sin θ, so segment = ½r²(θ − sin θ).

• What happens when the angle is 360°?

  At 360°, the sector becomes the full circle: area = πr², arc = 2πr, chord = 0 (the endpoints coincide), and segment area = 0.