Two concentric circles have radii $9 cm$ and $4 cm$ . Find (i) the area of the ring formed between them, and (ii) the length of the arc of the outer circle corresponding to a central angle of $3 0^{\circ}$ .

medium 3 marks Class 10 · Mathematics · Areas Related to Circles

Answer

(i)

65 π cm^{2}

, (ii)

\frac{3 π}{2} cm

The correct answer is (i) $65 π cm^{2}$ , (ii) $\frac{3 π}{2} cm$ — Here $θ = 3 0^{\circ}$ and $R = 9$ : $l = \frac{30}{360} \times 2 π \times 9 = \frac{1}{12} \times 18 π = \frac{3 π}{2} cm$ .

Step-by-step solution

The area of a ring (annulus) is $π (R^{2} - r^{2})$ where $R$ and $r$ are the outer and inner radii respectively.
Substitute $R = 9$ and $r = 4$ : $Area = π (9^{2} - 4^{2}) = π (81 - 16) = 65 π cm^{2}$ .
Arc length of the outer circle: $l = \frac{θ}{36 0 ^{\circ}} \times 2 π R$ .
Here $θ = 3 0^{\circ}$ and $R = 9$ : $l = \frac{30}{360} \times 2 π \times 9 = \frac{1}{12} \times 18 π = \frac{3 π}{2} cm$ .

Common mistake: Subtracting radii instead of their squares for the ring area

Concept tested: Area of ring and arc length of a circle

Published 21 August 2026 · How this solution was prepared and checked

Two concentric circles have radii 9 cm and 4 cm. Find (i) the area of the ring formed between them, and (ii) the length of the arc of the outer circle corresponding to a central angle of 30∘.

Step-by-step solution

Related questions

Two concentric circles have radii $9 cm$ and $4 cm$ . Find (i) the area of the ring formed between them, and (ii) the length of the arc of the outer circle corresponding to a central angle of $3 0^{\circ}$ .