How Do You Find the Length of an Arc and the Area of a Sector?

For IB Mathematics: Applications and Interpretation SL, find both quantities by multiplying the corresponding whole-circle measurement by the fraction $\frac{\theta}{360^\circ}$, where $\theta$ is the central angle in degrees.

This is SL 3.4: The circle, arc and area of sector, which uses degrees only.

The Reasoning

A sector with central angle $\theta$ occupies the fraction

$$\frac{\theta}{360^\circ}$$

of a complete circle. Therefore, multiply the circumference or area of the complete circle by this fraction.

Here, $r$ is the radius and $\theta$ is the angle at the centre.

Use this process in every question:

1. Identify the radius and the angle at the centre.

2. Decide whether the question asks for a curved length or a two-dimensional area.

3. Substitute into the appropriate degree formula.

4. Evaluate with the GDC, retaining full precision.

5. Round only the final value and attach the correct units.

If the diameter is given, first calculate $r=\frac{d}{2}$. Do not substitute the diameter as $r$.

Worked Example

A sector has radius $8\text{ cm}$ and central angle $60^\circ$.

For the arc length:

$$s=\frac{60^\circ}{360^\circ}\times 2\pi(8)=\frac{8\pi}{3}\approx 8.38\text{ cm}$$

For the sector area:

$$A=\frac{60^\circ}{360^\circ}\times \pi(8)^2=\frac{32\pi}{3}\approx 33.5\text{ cm}^2$$

The units differ because arc length is one-dimensional, while area is two-dimensional.

A reasonableness check supports both answers. Since $60^\circ$ is one sixth of a full turn, the arc should be one sixth of the circumference and the sector should be one sixth of the circle's area. Both exact results satisfy this check.

A common misconception is to use the radian formulas $s=r\theta$ and $A=\frac{1}{2}r^2\theta$. These formulas require to be in radians and belong to AHL 3.7, not SL 3.4. At SL, use the degree formulas containing .

Exam Technique

On any GDC-active Math AI paper, write the correct formula and substitution before giving the calculator value. Keep full precision until the end, give an exact answer or round to three significant figures unless instructed otherwise, and include $\text{cm}$ for arc length but $\text{cm}^2$ for sector area.