Arc, chord and sector at common angles
The table below illustrates how arc length compares with chord length for a circle of radius 10 at several common central angles.
| Central angle | Arc length (r=10) | Chord length (r=10) |
|---|---|---|
| 30° | 5.2360 | 5.1764 |
| 60° | 10.4720 | 10.0000 |
| 90° | 15.7080 | 14.1421 |
| 180° | 31.4159 | 20.0000 |
| 360° | 62.8319 (full circumference) | 0 (start = end point) |
- At 360°, the arc length equals the full circumference (2πr) and the chord length is zero because the start and end points coincide.
- Arc length is always greater than or equal to chord length for the same angle, since a straight line is the shortest path between two points and the arc curves between them.
- Segment area is only meaningful for angles strictly between 0° and 360°; at exactly 0° both sector and segment area are zero.
What are arc length, chord length and sector area?
An arc is a curved portion of a circle's circumference, bounded by two radii and a central angle. Arc length is the distance along that curve. A chord is the straight line segment connecting the arc's two endpoints — always shorter than the arc itself unless the angle is vanishingly small.
A sector is the pie-slice-shaped region bounded by the arc and the two radii; a segment is the smaller region bounded by the arc and the chord (the pie slice minus the triangle formed by the two radii and the chord). All four quantities are calculated directly from the radius and central angle.
How to use this arc length calculator
- Enter the circle's radius.
- Enter the central angle in degrees, between 0° and 360°.
- Read the arc length, chord length, sector area and segment area, which update instantly.
The formulas behind arc length and sector area
All formulas require the central angle in radians (θ = angle in degrees × π / 180); the calculator performs this conversion automatically from the degree input.
Common mistakes
- Entering the angle in radians when the calculator expects degrees.
- Confusing arc length (a curved distance) with chord length (a straight-line distance) — they are equal only in the limit as the angle approaches zero.
- Confusing sector area (bounded by two radii and the arc) with segment area (bounded by the chord and the arc) — the segment is always smaller.
- Using an angle greater than 360°, which does not correspond to a single, well-defined sector of one circle.
자주 묻는 질문
How do you calculate arc length?
Arc length equals radius times the central angle in radians: s = rθ. For a radius of 10 and a 60° angle (θ = π/3 radians), the arc length is 10 × π/3 ≈ 10.4720 units.
What is the difference between arc length and chord length?
Arc length is the distance along the curved edge of the circle between two points; chord length is the straight-line distance between the same two points. For radius 10 and a 60° angle, arc length is about 10.4720 and chord length is exactly 10 (since chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10).
How do you find the area of a sector?
Sector area equals half the radius squared times the central angle in radians: A = ½r²θ. For a radius of 10 and a 60° angle, the sector area is ½ × 100 × (π/3) ≈ 52.3599 square units.
What is a segment of a circle?
A segment is the region between a chord and the arc it cuts off — the sector area minus the triangular area formed by the two radii and the chord. For radius 10 and a 60° angle, the segment area is ½ × 100 × (π/3 − sin 60°) ≈ 9.0586 square units.
How do you convert degrees to radians for these formulas?
Multiply the angle in degrees by π/180. For example, 60° × π/180 = π/3 radians ≈ 1.0472 radians.
What is the arc length of a full circle?
At a central angle of 360° (2π radians), the arc length equals the full circumference, 2πr. For a radius of 10, that is 2π × 10 ≈ 62.8319 units.
참고 자료
- Weisstein, Eric W. "Circular Arc", "Circular Sector" and "Circular Segment." MathWorld — A Wolfram Web Resource.
- Standard trigonometry textbook conventions (e.g. Larson, Trigonometry, Cengage Learning).