Sector Area Calculator
Calculate sector area, segment area, arc length, chord, and perimeter
Sector Area Calculator
Understanding Sectors
A sector is a "pie slice" portion of a circle, bounded by two radii and an arc. The sector area depends on the radius and central angle. A segment is the region between the chord and the arc.
Key Terms:
- Sector: Region bounded by two radii and an arc
- Segment: Region between chord and arc
- Central Angle: Angle at the center of the circle
- Arc: Curved edge of the sector
Sector Formulas
How to Use This Sector Area Calculator
Pick the pair of values you know — radius and angle (in degrees or radians), sector area and radius, sector area and angle, or arc length and radius — enter them, and the calculator solves for everything else: sector area, segment area, arc length, chord length, perimeter, and what percentage of the full circle the sector covers.
Worked Example: A 60° Slice of a 12 cm Circle
Take a circle with radius r = 12 cm and a central angle of 60°. First convert the angle to radians: θ = 60 × π/180 ≈ 1.0472 rad. The sector area is A = ½r²θ = ½ × 144 × 1.0472 ≈ 75.4 cm² — exactly one sixth of the full circle's area (πr² ≈ 452.4 cm²), since 60° is one sixth of 360°. The arc length is s = rθ ≈ 12.57 cm, and the perimeter of the slice is 2r + s ≈ 36.57 cm. The chord across the slice is c = 2r sin(θ/2) = 2 × 12 × sin(30°) = 12 cm.
Degrees vs Radians
The compact formulas A = ½r²θ and s = rθ only work when θ is in radians. If you prefer degrees, use A = (θ/360)πr² instead — both give identical results. The calculator accepts either unit and reports the angle back in both, so you never need to convert by hand.
Where Sector Calculations Show Up
- Food and portioning: sizing pizza or cake slices by area rather than guesswork.
- Construction and landscaping: material estimates for fan-shaped patios, curved decks, and irrigation coverage zones.
- Engineering: cross-sections of cams, gears, and pie-shaped brackets.
- Data visualization: computing the angles behind pie-chart wedges from percentage shares.
Frequently Asked Questions
How do I calculate the area of a sector?+
Multiply half the radius squared by the central angle in radians: A = ½r²θ. If your angle is in degrees, use A = (θ/360)πr² instead. For example, a 90° sector of a circle with radius 10 has area (90/360) × π × 100 ≈ 78.54 square units — a quarter of the full circle.
What is the difference between a sector and a segment?+
A sector is the pie-slice region bounded by two radii and the arc between them. A segment is the smaller region between the chord and the arc — a sector with the central triangle cut away. Its area is ½r²(θ − sin θ), and this calculator reports both values at once.
Can I find the radius if I only know the sector area and angle?+
Yes. Choose the area-and-angle input mode and the calculator rearranges A = ½r²θ to solve r = √(2A/θ). It also works backwards from area and radius to find the angle, or from arc length and radius.
What is the perimeter of a sector?+
The perimeter is the two straight radii plus the curved arc: P = 2r + rθ with θ in radians. For a 60° sector of radius 12, that is 24 + 12.57 ≈ 36.57 units. This is useful for edging material around fan-shaped garden beds or trims.
Is this sector area calculator free and does it work on mobile?+
Yes. It is free, requires no sign-up, and runs entirely in your browser with no data sent to a server. The responsive layout works on phones, tablets, and desktops, so it is handy for homework and on-site measurements alike.
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