How do you calculate arc length?

Arc length S = R x theta where theta is in radians. If angle is in degrees: S = R x (deg x pi/180). Example: R=10, theta=60: S = 10 x pi/3 = 10.472. Arc length is proportional to both the radius and the central angle.

How do you calculate sector area?

Sector area A = (R^2 x theta)/2 where theta is in radians. In degrees: A = (pi x R^2 x deg)/360. Example: R=10, theta=60: A = (100 x pi/3)/2 = 52.36. The sector is like a slice of pie.

What is the difference between sector and segment?

A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. The segment area = sector area - triangle area (the triangle formed by the two radii and the chord). For small angles the segment is small; for large angles they are nearly equal.

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Home›Geometry Tools›Arc Length & Sector Area Calculator

Arc Length & Sector Area Calculator

Enter radius and central angle to compute arc length, sector area, and chord properties

Radius (R)

Central Angle (degrees)

Arc Length & Sector Area Formulas

Arc: S = R x theta (theta in radians)

Sector: A = (R^2 x theta) / 2

Chord: L = 2R x sin(theta/2)

Sagitta: h = R x (1 - cos(theta/2))

Converting: deg x pi/180 = rad

A circular sector is the region of a circle bounded by two radii and the arc between them. These formulas are essential in geometry, engineering, and design.

⚠ Enter the central angle in degrees (0-360). At 360 degrees the sector is the full circle: arc length = circumference, area = pi x R^2.

What Is a Circular Sector?

A circular sector is the part of a circle bounded by two radii and their connecting arc. The arc length is the curved boundary of the sector. The sector area represents the space enclosed by the two radii and the arc.

Arc Length

The curved distance along the circle edge. Proportional to the central angle. Full circle arc = 2piR.

Sector Area

The area of the pie slice. Half the product of R^2 and the angle in radians. Full circle = pi x R^2.

Chord

The straight line connecting the two endpoints of the arc. Always shorter than the arc for positive angles.

Segment vs Sector

Segment = sector - triangle (formed by radii and chord). Segment area is useful for calculating filled circular sections.

Teaching Example: Circle R=10, angle=60 deg. Radians = 60 x pi/180 = pi/3 = 1.047. Arc S = 10 x 1.047 = 10.472. Sector A = (100 x 1.047)/2 = 52.36. Chord L = 20 x sin(30) = 10. Sagitta h = 10 x (1 - cos(30)) = 1.34. Segment = 52.36 - (100 x sin(60))/2 = 52.36 - 43.30 = 9.06.

Applications

Engineering Architecture Pizza/Cutting Gear Design Trigonometry Construction

FAQs about Arc Length and Sector Area

How to calculate arc length?▼

S = R x theta (radians). S = R x (deg x pi/180). For R=10, deg=60: S = 10 x pi/3 = 10.47.

How to calculate sector area?▼

A = (R^2 x theta)/2 (radians) or A = (pi x R^2 x deg)/360. R=10, deg=60: A = 100 x pi/6 = 52.36.

What angle gives a half-circle sector?▼

180 degrees gives a half-circle (semicircle). Arc length = pi x R, sector area = (pi x R^2)/2, chord = 2R (the diameter).

What is the segment area?▼

Segment = sector area - triangle area. Triangle area = (R^2 x sin(theta))/2. For R=10, theta=60: segment = 52.36 - 43.30 = 9.06.

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