mathlumo.com
Reading progress0%

Current section：The central angle sets the share

Geometry

Arcs & Sectors

Slice a wedge off a circle — the arc takes a share of the circumference, the sector a share of the area, and the central angle sets the share.

A pizza arrives and you cut yourself a slice. Two things about that slice are worth measuring: how long the crust is, and how much pizza it covers. Each is a piece of a circle — an arc and a sector. And how big a piece? One number decides it: the central angle.

The central angle sets the share

The angle between two radii drawn from the center is the central angle. The whole circle is 360°360°, so an angle of nn hands you a natural fraction:

n360\frac{n}{360}

A 90°90° slice takes 90360=14\frac{90}{360} = \frac{1}{4} of the circle — one quarter of the crust, and one quarter of the pizza. Arc and sector always move together, and that sync rule is exactly why pie charts work.

InteractiveSector Cutter

How to use

rd

Diameter d

d = 6

Circumference C = 2πr

C = 18.85

Area A = πr²

A = 28.27

C = 2πr = 2π×3 ≈ 18.85 · A = πr² = π×3² ≈ 28.27

Use the explorer to dial in a radius and read off the full circumference and area. Then slice by the fraction from the central angle — the arc length and the sector area are both ready.

Two formulas: a share of each

L=n360×2πrS=n360×πr2L = \frac{n}{360} \times 2\pi r \qquad S = \frac{n}{360} \times \pi r^2

The formulas are twins with different foundations: arc length takes its share of the circumference, sector area takes its share of the area. Grabbing the wrong base is the most common slip.

Grab the right base

For a 60°60° arc of radius 3: 16×2π×3=π≈3.14\frac{1}{6} \times 2\pi \times 3 = \pi \approx 3.14. Take the area as the base by mistake and you get 16×9π≈4.71\frac{1}{6} \times 9\pi \approx 4.71 — about half too much.

Worked example: a 90° sector

Radius 6, central angle 90°90°:

  • fraction: 90360=14\frac{90}{360} = \frac{1}{4}
  • arc length L=14×2π×6=3π≈9.42L = \frac{1}{4} \times 2\pi \times 6 = 3\pi \approx 9.42
  • area S=14×π×62=9π≈28.27S = \frac{1}{4} \times \pi \times 6^2 = 9\pi \approx 28.27

The fair pizza question

Eight people share one pizza and every slice must be equal. What central angle? 360°÷8=45°360° \div 8 = 45°. Want double the crust? Cut a bigger angle — with the radius fixed, both arc and area answer only to the central angle.

Arcs and sectors around you

Pizza slices, fan blades, the region a clock hand sweeps, every wedge of a pie chart — all sectors. Comparing them takes one mantra: find the fraction first, then multiply by the whole. For more practice with fraction thinking, head to Percentages.

Check yourself

Quick quiz

0 / 3 correct0 / 3 correct

  1. 1. A 180° arc is what fraction of the circumference?

  2. 2. Radius 6, central angle 90°: what is the arc length?

  3. 3. The radius stays fixed but the central angle doubles. The sector area…