Circle Radius Calculation: Arc Length & Central Angle
The problem asks for the radius of a circle given the arc length and the central angle subtended by that arc. We are given:
- Arc length ($s$) = 2 cm
- Central angle ($\theta$) = $30^\circ$
- Value of $\pi$ = $\frac{22}{7}$
Convert Angle to Radians
The formula relating arc length, radius, and central angle requires the angle to be in radians. To convert degrees to radians, we use the conversion factor $\frac{\pi}{180^\circ}$.
$ \theta_{\text{rad}} = 30^\circ \times \frac{\pi}{180^\circ} = \frac{30\pi}{180} = \frac{\pi}{6} \text{ radians} $
Arc Length Formula
The formula for the arc length ($s$) of a circle is:
$ s = r \theta_{\text{rad}} $
Where '$r$' is the radius and '$\theta_{\text{rad}}$' is the central angle in radians.
Calculate Radius
Now, we substitute the known values into the formula and solve for the radius ($r$):
$ 2 \text{ cm} = r \times \frac{\pi}{6} $
Rearranging the formula to solve for $r$:
$ r = \frac{2 \text{ cm}}{\frac{\pi}{6}} = \frac{2 \times 6}{\pi} \text{ cm} = \frac{12}{\pi} \text{ cm} $
Substitute Pi Value
Using the given value $\pi = \frac{22}{7}$:
$ r = \frac{12}{\frac{22}{7}} \text{ cm} = 12 \times \frac{7}{22} \text{ cm} = \frac{84}{22} \text{ cm} $
Simplify and Final Answer
Simplify the fraction and convert it to a decimal:
$ r = \frac{84 \div 2}{22 \div 2} \text{ cm} = \frac{42}{11} \text{ cm} $
$ r \approx 3.818181... \text{ cm} $
Rounding to two decimal places, the radius is approximately 3.82 cm.
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