Arc Length and Sector Area: Pie Slices, Curved Fences, and Garden Beds
October 29, 2026
A sector is a pie-slice of a circle, bounded by two radii and an arc. If the central angle is θ degrees, the sector is a fraction θ/360 of the whole circle. So the arc length is (θ ÷ 360) × 2πr, and the area is (θ ÷ 360) × πr². With θ in radians, the formulas are simpler: arc length = rθ and area = ½r²θ.
For a circle with radius 10 feet and a central angle of 72°, the arc length is (72 ÷ 360) × 2π × 10 = 0.2 × 62.83 = 12.57 feet, and the area is 0.2 × π × 100 = 62.83 square feet. A 72° sector is exactly one fifth of the circle.
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Using degrees: arc length s = 2πr × θ/360; sector area A = πr² × θ/360. Using radians: s = rθ; A = ½ r²θ. To convert, 72° = 72 × π/180 = 1.2566 radians. Then s = 10 × 1.2566 = 12.57 and A = ½ × 100 × 1.2566 = 62.83, matching. The chord (the straight line between the arc's endpoints) is 2r sin(θ/2) = 2 × 10 × sin 36° = 11.76 feet. The area of the segment (the region between chord and arc) equals the sector area minus the triangle area: 62.83 − ½r² sin θ = 62.83 − 47.55 = 15.28 square feet.
| Central angle | Fraction of circle | Arc length (r = 10) | Sector area (r = 10) |
|---|---|---|---|
| 30° | 1/12 | 5.24 | 26.18 |
| 45° | 1/8 | 7.85 | 39.27 |
| 72° | 1/5 | 12.57 | 62.83 |
| 90° | 1/4 | 15.71 | 78.54 |
| 120° | 1/3 | 20.94 | 104.72 |
| 180° | 1/2 | 31.42 | 157.08 |
Worked Example: Curved Garden Edging
A fan-shaped flower bed is bounded by a curved edge with radius 12 feet and a central angle of 100°. The curved edge needs flexible edging of length (100 ÷ 360) × 2π × 12 = 20.94 feet. The straight sides total 24 feet. The bed area is (100 ÷ 360) × π × 144 = 125.7 square feet. At 3 inches of mulch (0.25 foot), you need 31.4 cubic feet, about 1.2 cubic yards. Using these formulas avoids the guesswork of measuring a curve with a tape and string.
Other Uses
Pizza and cake slices: a 12-inch pizza cut into 8 slices has each slice a 45° sector with area 14.1 square inches and a crust arc of 4.7 inches. Wheels and gears: a wheel of radius 14 inches that rotates through 90° moves its rim 22 inches. Clock hands: the minute hand of length 6 inches sweeps 6 degrees per minute, tracing an arc of 0.63 inches each minute at the tip. Navigation: one minute of latitude on Earth is a nautical mile because the arc length for 1/60 of a degree at a radius of 3,440 nautical miles is r × θ = 3,440 × 0.000291 = 1.0 nautical mile. Curved roads and railway tracks are laid out as arcs of given radius.
Common Mistakes
Using a diameter instead of a radius; mixing degrees with radian formulas; forgetting to add the straight edges when computing a perimeter; and treating a sector as a triangle. Another is rounding too early: carry π to enough digits. Check your answers using proportions: a 90° sector must be a quarter of the circle's circumference and area.
Frequently Asked Questions
What is the arc length formula?
Arc length = (θ ÷ 360) × 2πr in degrees, or rθ in radians.
What is the area of a sector?
(θ ÷ 360) × πr² in degrees, or ½r²θ in radians.
What is a circular segment?
The region between a chord and its arc, equal to the sector area minus the triangle area.
How do I convert degrees to radians?
Multiply by π and divide by 180.
How long is the arc for 90 degrees and radius 10?
15.71 units.