Unless stated otherwise, use the approximation for . 3. Calculate the length of the arc of a circle if: (i) the radius is and the angle at the centre is , and (ii) the radius is and the angle at the centre is .

Understand the Question The length of an arc of a circle of radius subtending an angle at the centre is a fraction of the total circumference: We substitute the given values of radius , central angle , and to find the arc length for each part. Part (i) The radius is and the angle at the centre is . Step 1 · Calculate the Arc Length Given radius and central angle .Using the formula Answer > (i) (or ) Part (ii) The radius is and the angle at the centre is . Step 1 · Calculate the Arc Length Given radius and central angle .Using the formula Answer > (ii) Common Mistakes Formula Confusion: Confusing the arc length formula with the sector area formula . Incorrect Units: Forgetting to append the correct units ( in part (i) and in part (ii)) to the final answers. Decimal Conversion Errors: Miscalculating when working directly with decimals like or instead of converting them to simple fractions ( or ) to easily cancel out factors with .

Measuring Space: Perimeter and Area | Exercise 6.1

Question 3

Unless stated otherwise, use the approximation $\dfrac{22}{7}$ for $\pi$ .

Calculate the length of the arc of a circle if:

(i) the radius is $3.5\text{ cm}$ and the angle at the centre is $60^\circ$ , and

(ii) the radius is $6.3\text{ m}$ and the angle at the centre is $120^\circ$ .

Solution

Understand the Question

The length of an arc of a circle of radius $r$ subtending an angle $heta$ at the centre is a fraction of the total circumference: $L = \dfrac{\theta}{360^\circ} \times 2\pi r$
We substitute the given values of radius $r$ , central angle $heta$ , and $\pi = \dfrac{22}{7}$ to find the arc length for each part.

(i) The radius is $3.5\text{ cm}$ and the angle at the centre is $60^\circ$ .

Step 1 · Calculate the Arc Length

Given radius $r = 3.5\text{ cm} = \dfrac{7}{2}\text{ cm}$ and central angle $\theta = 60^\circ$ .Using the formula $L = \dfrac{\theta}{360^\circ} \times 2\pi r$

\begin{aligned} L &= \dfrac{60}{360} \times 2 \times \dfrac{22}{7} \times 3.5 \\[0.6em] &= \dfrac{1}{6} \times 2 \times \dfrac{22}{7} \times \dfrac{7}{2} \\[0.6em] &= \dfrac{1}{6} \times 22 \\[0.6em] &= \dfrac{11}{3} \\[0.6em] &= 3.666\dots \approx 3.67\text{ cm} \end{aligned}

Answer

(i) $3.67\text{ cm}$ (or $\dfrac{11}{3}\text{ cm}$ )

(ii) The radius is $6.3\text{ m}$ and the angle at the centre is $120^\circ$ .

Step 1 · Calculate the Arc Length

Given radius $r = 6.3\text{ m} = \dfrac{63}{10}\text{ m}$ and central angle $\theta = 120^\circ$ .Using the formula $L = \dfrac{\theta}{360^\circ} \times 2\pi r$

\begin{aligned} L &= \dfrac{120}{360} \times 2 \times \dfrac{22}{7} \times 6.3 \\[0.6em] &= \dfrac{1}{3} \times 2 \times \dfrac{22}{7} \times \dfrac{63}{10} \\[0.6em] &= \dfrac{1}{3} \times 2 \times 22 \times \dfrac{9}{10} \\[0.6em] &= \dfrac{1}{3} \times 44 \times \dfrac{9}{10} \\[0.6em] &= 44 \times \dfrac{3}{10} \\[0.6em] &= \dfrac{132}{10} \\[0.6em] &= 13.2\text{ m} \end{aligned}

Answer

(ii) $13.2\text{ m}$

Common Mistakes

Formula Confusion: Confusing the arc length formula $L = \dfrac{\theta}{360^\circ} \times 2\pi r$ with the sector area formula $A = \dfrac{\theta}{360^\circ} \times \pi r^2$ .
Incorrect Units: Forgetting to append the correct units ( $\text{cm}$ in part (i) and $\text{m}$ in part (ii)) to the final answers.
Decimal Conversion Errors: Miscalculating when working directly with decimals like $3.5$ or $6.3$ instead of converting them to simple fractions ( $\dfrac{7}{2}$ or $\dfrac{63}{10}$ ) to easily cancel out factors with $\pi = \dfrac{22}{7}$ .

Question 3

More questions in Exercise 6.1