Arcs and Sectors

Arcs and Sectors

An arc is a portion of the circumference; a sector is the region bounded by two radii and an arc. Both are calculated as a fraction of the full circle, where the fraction is determined by the sector angle.

Formulae

\[ \text{Arc length} = \frac{\theta}{360}\times 2\pi r \qquad \text{Sector area} = \frac{\theta}{360}\times \pi r^2 \]

where \( \theta \) is the sector angle in degrees and \( r \) is the radius.

Segment Area

A segment is bounded by a chord and an arc. Its area is found by subtracting the triangle from the sector:

\[ \text{Segment area} = \text{Sector area} - \text{Triangle area} = \frac{\theta}{360}\pi r^2 - \frac{1}{2}r^2\sin\theta \]

Finding the Angle from Arc Length or Area

Rearrange the formulae:

\[ \theta = \frac{\text{arc length}}{2\pi r}\times360 \qquad \theta = \frac{\text{sector area}}{\pi r^2}\times360 \]

Perimeter of a Sector

The perimeter of a sector includes two radii and the arc:

\[ P = 2r + \frac{\theta}{360}\times2\pi r \]

Worked Examples

Sector: radius 9 cm, angle 80°. Find the arc length and sector area.
\[ \text{Arc} = \frac{80}{360}\times2\pi\times9 = 4\pi\approx12.57 \text{ cm} \] \[ \text{Area} = \frac{80}{360}\times\pi\times81 = 18\pi\approx56.55 \text{ cm}^2 \]
A sector has arc length \( 10\pi \) cm and radius 15 cm. Find the sector angle.
\[ \theta = \frac{10\pi}{2\pi\times15}\times360 = \frac{10}{30}\times360 = 120° \]
Find the area of the minor segment in a circle of radius 8 cm with a chord subtending 60° at the centre.
\[ \text{Sector area} = \frac{60}{360}\pi\times64 = \frac{32\pi}{3} \] \[ \text{Triangle area} = \frac{1}{2}\times64\times\sin60° = 32\times\frac{\sqrt{3}}{2} = 16\sqrt{3} \] \[ \text{Segment} = \frac{32\pi}{3} - 16\sqrt{3}\approx33.51-27.71\approx5.8 \text{ cm}^2 \]

 Key Takeaways

  • Arc length: \( \frac{\theta}{360}\times2\pi r \). Sector area: \( \frac{\theta}{360}\times\pi r^2 \).
  • Perimeter of sector = arc + 2 radii.
  • Segment area = sector area − triangle area.
  • To find \( \theta \) from arc or area: rearrange the formula.
  • The fraction \( \theta/360 \) is the key multiplier — it scales the full-circle formula to the partial sector.