QUESTION 3
Easy
Calculate the length of the arc of a circle if:
(i) the radius is 3.5 cm and the angle at the centre is 60°.
(ii) the radius is 6.3 cm and the angle at the centre is 120°.
SOLUTION
1
(i) \(r = 3.5\,cm,\,θ = 60°\)
Using arc length formula
\(L = \frac{θ}{360} × 2πr\) \(\begin{aligned} L &= \frac{θ}{360} × 2πr \\ &= \frac{60}{360} × 2 × \frac{22}{7} × 3.5 \\ &= \frac{1}{6} × 2 × 11 \\ &= \frac{22}{6} \\ &= \frac{11}{3} \\ &= 3.67 \,cm\end{aligned}\)
The length of the arc of the circle is 3.67 cm.
\(L = \frac{θ}{360} × 2πr\) \(\begin{aligned} L &= \frac{θ}{360} × 2πr \\ &= \frac{60}{360} × 2 × \frac{22}{7} × 3.5 \\ &= \frac{1}{6} × 2 × 11 \\ &= \frac{22}{6} \\ &= \frac{11}{3} \\ &= 3.67 \,cm\end{aligned}\)
The length of the arc of the circle is 3.67 cm.
2
(i) \(r = 6.3\,cm,\,θ = 120°\)
Using arc length formula
\(L = \frac{θ}{360} × 2πr\) \(\begin{aligned} L &= \frac{θ}{360} × 2πr \\ &= \frac{120}{360} × 2 × \frac{22}{7} × 6.3 \\ &= \frac{1}{3} × \frac{44}{7} × \frac{63}{10} \\ &= \frac{1}{3} × 44 × \frac{9}{10} \\ &= \frac{396}{30} \\ &= 13.2 \,cm\end{aligned}\)
The length of the arc of the circle is 13.2 cm.
\(L = \frac{θ}{360} × 2πr\) \(\begin{aligned} L &= \frac{θ}{360} × 2πr \\ &= \frac{120}{360} × 2 × \frac{22}{7} × 6.3 \\ &= \frac{1}{3} × \frac{44}{7} × \frac{63}{10} \\ &= \frac{1}{3} × 44 × \frac{9}{10} \\ &= \frac{396}{30} \\ &= 13.2 \,cm\end{aligned}\)
The length of the arc of the circle is 13.2 cm.
Concept Note
An arc is part of the circumference, given by the formula
\(Arc length, L = \frac{θ}{360} × 2πr\)
where
\(θ =\) central angle
\(r =\) radius