Question

An arc of a circle of radius 21 cm subtends an angle of \( 60^\circ \) at the centre. Find the area of the sector formed by the arc.

Hint:

For sector problems, always use the proportional relationship: \[ \frac{\text{Angle at centre}}{360^\circ} = \frac{\text{Area of sector}}{\text{Area of circle}}. \] This simplifies calculations for any given central angle.

Answer 1

Step 1: Formula for area of a sector.
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta = 60^\circ \) and \( r = 21 \, \text{cm} \).
Step 2: Substitute values.
\[ \text{Area} = \frac{60}{360} \times \pi \times (21)^2 \] \[ = \frac{1}{6} \times \pi \times 441 \] \[ = 73.5\pi \]
Step 3: Express in numerical form.
Taking \( \pi = 3.14 \): \[ \text{Area} = 73.5 \times 3.14 = 230.79 \, \text{cm}^2 \]
Step 4: Final Answer.
\[ \boxed{\text{Area of the sector} = 73.5\pi \, \text{cm}^2 = 230.79 \, \text{cm}^2} \]