Find the angle subtended by an arc 33cm long of a circle of diameter 28cm.

Analysis

To find the angle subtended by an arc at the center of a circle, we use the formula for arc length: Arc Length ($L$) = $\frac{\theta}{360^\circ} \times 2\pi r$ Where:

• $L$ = length of the arc
• $\theta$ = angle subtended at the center
• $r$ = radius of the circle
• $\pi$ = approximately $\frac{22}{7}$

Concept

The arc length is a fraction of the total circumference of the circle. Since the total circumference is $2\pi r$ (or $\pi d$), the arc length is simply the ratio of the angle to the full circle ($360^\circ$) multiplied by the circumference.

Step-by-Step Solution

1. Identify the given values:

• Arc length ($L$) = $33\text{ cm}$
• Diameter ($d$) = $28\text{ cm}$
• Radius ($r$) = $\frac{d}{2} = \frac{28}{2} = 14\text{ cm}$

2. Set up the equation: Using the formula $L = \frac{\theta}{360^\circ} \times 2\pi r$: $33 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times 14$

3. Simplify the right side:

• First, simplify $2 \times \frac{22}{7} \times 14$: $2 \times 22 \times 2 = 88$
• Now the equation is: $33 = \frac{\theta}{360} \times 88$

4. Solve for $\theta$:

• Multiply both sides by 360: $33 \times 360 = \theta \times 88$
• $11,880 = 88\theta$
• $\theta = \frac{11,880}{88}$
• $\theta = 135^\circ$

Conclusion

The calculation confirms that the angle subtended by the arc is $135^\circ$.

Exam Tip

When dealing with circles in JAMB, always check if you are given the diameter or the radius. A common mistake is using the diameter directly in the $2\pi r$ formula. Remember: $2r = d$, so you can also write the formula as $L = \frac{\theta}{360} \times \pi d$. Using this shortcut, $33 = \frac{\theta}{360} \times \frac{22}{7} \times 28$, which simplifies to $33 = \frac{\theta}{360} \times 88$, saving you time!