Angle Calculation Using Arc Length and Radius

The problem asks us to find the measure of an angle in degrees, given the radius of the circle and the length of the arc it subtends.

Information Given:

• Radius of the circle, $r$ = 28 cm
• Arc length, $l$ = 22 cm

Formula Used:

The relationship between arc length ($l$), radius ($r$), and the angle ($\theta$) in radians subtended at the centre is given by:

$l = r\theta$

To find the angle in degrees, we first find the angle in radians and then convert it.

Step 1: Calculate the angle in radians

Rearranging the formula, we get:

$\theta = \frac{l}{r}$

Substituting the given values:

$\theta = \frac{22 \text{ cm}}{28 \text{ cm}} = \frac{11}{14} \text{ radians}$

Step 2: Convert radians to degrees

The conversion factor from radians to degrees is $\frac{180^{\circ}}{\pi}$.

Angle in degrees = Angle in radians $\times \frac{180^{\circ}}{\pi}$

Angle in degrees = $\frac{11}{14} \times \frac{180^{\circ}}{\pi}$

We can approximate $\pi$ as $\frac{22}{7}$.

Angle in degrees = $\frac{11}{14} \times \frac{180^{\circ}}{\frac{22}{7}}$

Angle in degrees = $\frac{11}{14} \times \frac{180^{\circ} \times 7}{22}$

Simplify the expression:

Angle in degrees = $\frac{11}{2 \times 7} \times \frac{180^{\circ} \times 7}{2 \times 11}$

Cancel out common terms (11 and 7):

Angle in degrees = $\frac{1}{2} \times \frac{180^{\circ}}{2}$

Angle in degrees = $\frac{180^{\circ}}{4}$

Angle in degrees = $45^{\circ}$

Final Answer:

The degree measure of the angle subtended at the centre of the circle is $45^{\circ}$.