Q.Calculate the length of the arc of a circle of radius 31.0 cm which subtends an angle of $\dfrac{\pi}{6}$ at the centre.

The key idea is that arc length is directly proportional to the angle subtended at the centre, using the formula $l = r \theta$ (where $\theta$ is in radians).

Step 1: Identify the given values.

Radius $r = 31.0$ cm, angle $\theta = \dfrac{\pi}{6}$ rad.

Step 2: Apply the arc length formula.

$l = r \theta = 31.0 \times \dfrac{\pi}{6}$

Step 3: Compute numerically. …

The key idea here is that arc length is a direct proportion of the radius and the angle. When the angle is given in radians, the formula is beautifully simple: $s = r\theta$. This works because one radian is defined as the angle that subtends an arc equal in length to the radius. So if you have $\theta$ radians, you're essentially stacking $\theta$ such radius-length arcs together.

Let's walk through it step by step.

1. Identify what's given.

   The radius $r = 31.0$ cm. The central angle $\theta = \dfrac{\pi}{6}$ radians. Notice the angle is already in radians — that's important. If it were in degrees, we'd have to convert first.

2. Recall the arc length formula.

   For a circle, the length $s$ of an arc subtended by an angle $\theta$ (in radians) at the centre is:

This is not a coincidence — it's the definition of the radian measure.

3. Plug in the values.

4. Simplify the expression.

5. Get a numerical value (if needed). Using $\pi \approx 3.1416$:

Rounding to three significant figures (since 31.0 has three), we get $s \approx 16.2$ cm. …

Concept: Arc Length of a Circle

The length of an arc is directly proportional to the angle it subtends at the centre.

If the angle is given in radians, the formula is clean and simple.

Method: Direct Formula for Arc Length (Angle in Radians)

Formula:

Where:

• $l$ = arc length
• $r$ = radius of the circle
• $\theta$ = angle subtended at the centre (in radians)

Steps

1. Identify the given values

   • Radius, $r = 31.0 \, \text{cm}$
   • Angle, $\theta = \dfrac{\pi}{6}$ radians

2. Apply the formula

3. Simplify

4. Numerical approximation (if needed) Using $\pi \approx 3.1416$: …

Here’s a breakdown of the common mistakes students make when solving this problem, along with how to avoid each one.

1. Using the wrong formula for arc length

Mistake:

Students often confuse the formula for arc length with the formula for the area of a sector.

• Wrong: $A = \frac{1}{2} r^2 \theta$ (this is area, not length).
• Wrong: $l = \frac{\theta}{360^\circ} \times 2\pi r$ without checking if $\theta$ is in degrees or radians.

How to avoid:

Always recall the definition:

Arc length $l = r \theta$, where $\theta$ is in radians.

If the angle is given in degrees, convert first:

$\theta_{\text{rad}} = \frac{\pi}{180^\circ} \times \theta_{\text{deg}}$.

Here, $\theta = \frac{\pi}{6}$ is already in radians, so directly use:

2. Forgetting to keep units consistent

Mistake:

Using radius in cm but angle in degrees without conversion, or mixing cm and m.

How to avoid:

• The formula $l = r\theta$ requires $\theta$ in radians (a pure number).
• The unit of $l$ will be the same as the unit of $r$.
• Here $r = 31.0\ \text{cm}$, so $l$ will be in cm. No conversion needed.

3. Incorrect simplification of $\pi$ or rounding too early

Mistake:

• Using $\pi = 3.14$ too early leads to rounding errors.
• Or writing $l = 31.0 \times \frac{22}{7 \times 6}$ and getting a messy fraction.

How to avoid:

Keep $\pi$ symbolic until the final step. Only substitute a numerical value (like $3.14$ or $\frac{22}{7}$) at the end, and round as per the problem’s requirement.

Correct approach:

If a decimal is needed:

4. Misreading the angle — thinking it’s in degrees

Mistake:

Seeing $\frac{\pi}{6}$ and treating it as $30^\circ$ but then using the degree-based formula incorrectly. …