A piece of wire 20 cm long is bent into an arc of a circle subtending an angle of \(60^\circ\) at the centre. Find the radius of the circle.
\(\dfrac{60}{\pi}\;\text{cm}\) (≈ 19.10 cm)
Step 1: Recall the formula for arc length of a circle:
\( s = r \theta \)
where,
Step 2: Write down the given values:
Step 3: Convert the angle into radians, because the formula works in radians.
\(60^\circ = \dfrac{60 \times \pi}{180} = \dfrac{\pi}{3}\,\text{rad}\)
Step 4: Substitute the values into the formula:
\( s = r \theta \)
\( 20 = r \times \dfrac{\pi}{3} \)
Step 5: Solve for \(r\):
\( r = \dfrac{20}{\pi/3} = \dfrac{20 \times 3}{\pi} = \dfrac{60}{\pi}\,\text{cm} \)
Step 6: Approximate the value using \(\pi \approx 3.14\):
\( r \approx \dfrac{60}{3.14} = 19.10\,\text{cm} \)
Final Answer: The radius of the circle is \(\dfrac{60}{\pi}\,\text{cm}\) or about 19.10 cm.