Question

A sector of a circle makes a 127° angle at its centre. If the arc of the sector has length 36 mm, find

the perimeter of the sector.

Answer 1

Approximately
`68.5\; \rm mm`.

Explanation:

Convert the angle of this sector to radians:

`\begin{aligned}\theta &= 127^(\circ) \\ &= 127^(\circ) * (2\pi)/(360^(\circ)) \\ &\approx 2.22\end{aligned}`.

The formula
`s = r\, \theta` relates the arc length
`s` of a sector of angle
`\theta` (in radians) to the radius
`r` of this sector.

In this question, it is given that the arc length of this sector is
`s = 36\; \rm mm`. It was found that
`\theta = 2.22` radians. Rearrange the equation
`s = r\, \theta` to find the radius
`r` of this sector:

`\begin{aligned} r&= (s)/(\theta) \\ &\approx (36\; \rm mm)/(2.22) \\ &\approx 16.2\; \rm mm\end{aligned}`.

The perimeter of this sector would be:

`\begin{aligned}& 2\, r + s \\ =\; & 2 * 16.2\; {\rm mm} + 36\; {\rm mm} \\ =\; & 68.5\; \rm mm\end{aligned}`.