Finding Circle Radius from Arc Length and Central Angle

The problem asks for the radius of a circle given the length of an arc and the angle it subtends at the center.

Key Information

• Arc Length ($L$): $22\text{ cm}$
• Central Angle ($\theta$): $72^\circ$
• Goal: Find the Radius ($r$)

Arc Length Formula

The formula relating arc length, central angle (in degrees), and radius is:

$ L = \frac{\theta}{360^\circ} \times 2\pi r $

Calculation Steps

1. Substitute Known Values: Plug the given arc length and central angle into the formula.
   $ 22 = \frac{72^\circ}{360^\circ} \times 2\pi r $
2. Simplify the Fraction: Reduce the angle fraction.
   $ \frac{72}{360} = \frac{1}{5} $ So the equation becomes:
   $ 22 = \frac{1}{5} \times 2\pi r $ $ 22 = \frac{2\pi r}{5} $
3. Isolate Radius ($r$): Rearrange the equation to solve for $r$.
   Multiply both sides by 5:
   $ 22 \times 5 = 2\pi r $ $ 110 = 2\pi r $ Divide both sides by $2\pi$:
   $ r = \frac{110}{2\pi} $ $ r = \frac{55}{\pi} $
4. Substitute $\pi$ Value: Use the approximation $\pi \approx \frac{22}{7}$ for calculation.
   $ r = \frac{55}{\frac{22}{7}} $ $ r = \frac{55 \times 7}{22} $
5. Final Calculation: Simplify the expression.
   $ r = \frac{5 \times 11 \times 7}{2 \times 11} $ $ r = \frac{5 \times 7}{2} $ $ r = \frac{35}{2} $ $ r = 17.5 $

Result

The radius of the circle is $17.5\text{ cm}$.