Arc length and sector area

The parts of a circle

• An arc is a section of the circumference — part of the curved edge.
• A sector is the pie-slice region enclosed by two radii and an arc.
• A segment is the region cut off by a chord — the arc and the straight line joining its ends.

• The angle $\theta$ at the centre is what controls everything. Both formulae are simply "a fraction of the whole circle", and that fraction is $\dfrac{\theta}{360°}$.

Arc length

$$\ell = \frac{\theta}{360} \times 2\pi r$$

• Each part:

  • $\theta$ — the angle at the centre, in degrees
  • $r$ — the radius
  • $2\pi r$ — the full circumference
  • $\dfrac{\theta}{360}$ — the fraction of the circle the arc covers

• The logic is proportion. A $90°$ arc is a quarter of the way round, so it is a quarter of the circumference.

• Worked through — the arc of a circle of radius 12 cm subtending $70°$:

$$\ell = \frac{70}{360} \times 2\pi(12) = 0.19444 \times 75.398 = 14.66 \text{ cm}$$

Sector area

$$A = \frac{\theta}{360} \times \pi r^2$$

• The same fraction, applied to the area of the whole circle instead of its circumference.
• Worked through — the sector of a circle of radius 12 cm with angle $70°$:

$$A = \frac{70}{360} \times \pi(144) = 0.19444 \times 452.39 = 87.96 \text{ cm}^2$$

• Note the units. Arc length is in cm; sector area is in cm². Giving an area in centimetres, or a length in square centimetres, is an easy mark to lose.

Working backwards

• Both formulae can be rearranged for $\theta$ or $r$.
• For the angle:

$$\theta = \frac{360\ell}{2\pi r} \qquad \text{or} \qquad \theta = \frac{360A}{\pi r^2}$$

• For the radius from an arc:

$$r = \frac{360\ell}{2\pi\theta}$$

• Substitute what you know and solve rather than trying to memorise every rearrangement.

The perimeter of a sector

• The perimeter of a sector is not the arc alone. It is the arc plus the two radii:

$$P = \ell + 2r$$

• This is the single most common error in the topic. A question asking for "the perimeter of the sector" or "the length of edging around the garden bed" wants all three edges.
• Read the context. If a fence runs around a sector-shaped paddock, it follows all three sides; if it is only the curved boundary, it is the arc alone.

The area of a segment

• The segment area is the sector minus the triangle formed by the two radii and the chord:

$$A_{\text{segment}} = \underbrace{\frac{\theta}{360}\pi r^2}_{\text{sector}} - \underbrace{\frac{1}{2}r^2\sin\theta}_{\text{triangle}}$$

• The triangle uses the area formula from the previous page, with both sides equal to $r$ and the included angle $\theta$.
• This connects the two halves of the standard, and it is a typical Merit or Excellence question.

Radians — a forward reference

• At Level 3 these formulae are written in radians, where they become much simpler: $\ell = r\theta$ and $A = \frac{1}{2}r^2\theta$.
• At Level 2 you work in degrees, and your calculator must be in degree mode.