Circle Sector Calculator
Calculate the area, arc length, and perimeter of a circle sector or segment.
What Is the Circle Sector Calculator?
The Circle Sector Calculator computes the arc length, sector area, chord length, and perimeter of a circular sector given the radius and central angle. Enter the radius and angle (in degrees or radians) to get all measurements of the pie-slice shaped region.
Formula
How to Use
Enter the radius of the circle and the central angle of the sector. Choose degrees or radians. The calculator instantly computes arc length (curved boundary), sector area (the pie-slice area), chord length (straight line across the arc), and total perimeter (two radii + arc).
Example Calculation
Sector with r=10 cm, θ=60°: Arc = π×10×60/180 = 10π/3 ≈ 10.47 cm. Area = π×100×60/360 = 100π/6 ≈ 52.36 cm². Chord = 2×10×sin(30°) = 20×0.5 = 10 cm. Perimeter = 10+10+10.47 = 30.47 cm.
Understanding Circle Sector
A circle sector (or circular sector) is the region bounded by two radii of a circle and the arc between them, forming a shape resembling a slice of pie. The size of the sector is defined by its central angle — a full circle (360°) gives the entire disk, a half circle (180°) gives a semicircle, and a quarter circle (90°) gives a quadrant.
The arc length formula L = rθ (with θ in radians) is one of the most fundamental relationships in circular geometry — it defines what a radian is: the angle for which arc length equals the radius. The sector area A = ½r²θ follows from integrating the area swept by the radius as the angle varies.
Circle sector calculations appear in gear design (involute tooth geometry), pie chart creation, clock design (hand angles), irrigation pivot systems, antenna coverage areas, and architectural design of circular spaces. The chord length formula is particularly useful in surveying and construction, where the straight-line distance between two points on a circular boundary is needed.
Frequently Asked Questions
What is the difference between a sector and a segment?
A sector is the pie-slice region bounded by two radii and an arc. A segment is the region between a chord and an arc (sector minus the triangle formed by the two radii and chord).
How do I convert degrees to radians?
Radians = degrees × π/180. Key values: 360°=2π, 180°=π, 90°=π/2, 60°=π/3, 45°=π/4, 30°=π/6.
What is the formula for a semicircle (sector with 180°)?
Area = πr²/2, Arc = πr, Perimeter = πr + 2r = r(π+2).
What is a radian?
One radian is the angle subtended at the center by an arc equal in length to the radius. A full circle = 2π radians ≈ 6.283 radians. It is the natural angular unit for calculus.
Is this calculator free?
Yes, completely free with no registration required.
Related Tools
Circle Calculator — Area & Circumference
Calculate radius, diameter, circumference, and area of a circle from any known v…
Area Calculator — Shapes & Formulas
Calculate the area of rectangles, circles, triangles, and other common shapes.…
Arc Length Calculator — Curve Length
Calculate the arc length of a curve using numerical integration.…