In the given figure, \(O\) is the centre of circle. \(XYZ\) is an arc of the circle subtending an angle of \(45^{\circ}\) at the centre. If the radius of the circle is 32 cm, then the length of the arc \(XYZ\) is :
Show Hint
Remember that \(45^{\circ}\) is exactly \(\frac{1}{8}\) of a full circle (\(360^{\circ}\)). So the arc length is simply \(\frac{1}{8}\) of the circumference (\(2\pi r\)).
Step 1: Understanding the Concept:
The length of an arc is determined by the radius and the angle it subtends at the center. Step 2: Key Formula or Approach:
\[ \text{Arc Length} = \frac{\theta}{360^{\circ}} \times 2\pi r \] Step 3: Detailed Explanation:
Given \(\theta = 45^{\circ}\) and \(r = 32\) cm.
\[ \text{Length of arc } XYZ = \frac{45^{\circ}}{360^{\circ}} \times 2\pi(32) \]
Simplify the fraction: \( \frac{45}{360} = \frac{1}{8} \).
\[ \text{Length} = \frac{1}{8} \times 64\pi \]
\[ \text{Length} = 8\pi \text{ cm} \] Step 4: Final Answer:
The length of the arc \(XYZ\) is \(8\pi\) cm.
