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A circle has a diameter of 8 cm. Find the length of the arc subtended by a central angle of 45 degrees. Options: a) 2pi cm, b) pi cm, c) 4pi cm, d) 8pi cm

Mathematics

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Step-by-step solution

6 steps

  1. Identify the given values.

    The diameter of the circle is d=8d = 8 cm.
    The angle subtended at the center is θ=45∘\theta = 45^\circ.

  2. Calculate the radius of the circle.

    The radius rr is half of the diameter.

    r=d2=8 cm2=4 cmr = \frac{d}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm}

  3. Use the formula for the length of an arc.

    The length of an arc LL is given by the formula:

    L=θ360∘×2πrL = \frac{\theta}{360^\circ} \times 2\pi r

  4. Substitute the values into the formula.

    L=45∘360∘×2π(4 cm)L = \frac{45^\circ}{360^\circ} \times 2\pi (4 \text{ cm})

  5. Simplify the expression.

    L=18×8π cmL = \frac{1}{8} \times 8\pi \text{ cm} L=π cmL = \pi \text{ cm}

  6. Compare the result with the given options.

    The calculated arc length is π\pi cm. This matches option (b).

    The final answer is b) π\boxed{\text{b) } \pi}.

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