How a sector's measurements are found
A sector is the pie-slice piece of a circle cut by two radii and the arc between them. Its arc length and area are simply the full circle's circumference and area scaled down by the fraction of the full turn (θ ÷ 360°) that the central angle covers. The chord — the straight line joining the two ends of the arc — comes from the isosceles triangle formed by the two radii and follows a separate sine relationship.
- r — radius — center to edge
- θ — central angle in degrees (0° < θ < 360°)
- L — arc length
- A — sector area
More detail
Why perimeter includes the two radii
A sector's boundary isn't just the curved arc — it's a closed shape made of the arc plus the two straight radii that cut it from the circle, like a slice of pie including its two straight edges. That's why perimeter = arc length + 2r rather than just the arc length alone.
Chord vs. arc: two different distances
The chord is the straight-line shortcut between the arc's endpoints, while the arc is the curved path along the edge. They only match when θ is very small; as θ grows the arc curves away from the chord, and at θ = 180° the chord becomes a full diameter (2r) while the arc is a full semicircle (πr).
Angle in radians instead? If your angle is given in radians, convert first: degrees = radians × 180/π. This calculator expects degrees, so a quarter turn (π/2 rad) should be entered as 90.
Frequently asked questions
What is the area of a sector with radius 10 and angle 90°?
Area = π·r²·θ/360 = π·10²·90/360 = 25π ≈ 78.5398. That's exactly a quarter of the full circle's area (π·10² ≈ 314.159), since 90° is one quarter of 360°.
How is arc length different from sector perimeter?
Arc length is only the curved edge: L = 2πr·θ/360. Perimeter is the full boundary of the pie-slice shape — the arc plus the two straight radii: Perimeter = L + 2r. For r = 5, θ = 60° that's arc ≈ 5.236 but perimeter ≈ 15.236.
Why isn't the chord the same as the arc length?
The chord (2r·sin(θ/2)) is the straight line between the arc's two endpoints; the arc follows the curve. For a half-circle (θ = 180°) the chord equals the diameter (2r) while the arc equals half the circumference (πr) — clearly different numbers for the same sector.
Can the central angle be more than 360°?
No — 360° is a full circle, so a sector angle above that has no geometric meaning and this calculator returns no result. Enter a value strictly between 0° and 360°.