How Do You Calculate Arc Length and Sector Area Using Radians?

Use $s=r\theta$ for arc length and $A=\frac{1}{2}r^2\theta$ for sector area, where $r$ is the radius and $\theta$ is the central angle measured in radians. These formulas belong to IB Mathematics: Applications and Interpretation AHL 3.7: Radian measure, so they are HL-only content.

Why the formulas work

A complete circle contains $2\pi$ radians. Therefore, a sector with angle $\theta$ represents the fraction

$$\frac{\theta}{2\pi}$$

of the complete circle. Applying this fraction to the circumference and area gives:

The key idea is proportionality: doubling the central angle doubles both quantities when the radius is fixed. A radian is the angle subtending an arc equal in length to the radius, explaining the simple relationship $s=r\theta$.

If an angle is given in degrees, first convert it:

$$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\times\frac{\pi}{180}.$$

For example, $60^\circ=\frac{\pi}{3}$ radians. Do not substitute $60$ directly into a radian formula.

Worked example

A sector has radius $8\text{ cm}$ and central angle $1.2$ radians.

$$s=8(1.2)=9.6\text{ cm}$$ $$A=\frac{1}{2}(8^2)(1.2)=38.4\text{ cm}^2$$

Notice that arc length uses centimetres, whereas sector area uses square centimetres. The radius is squared only in the area formula, so checking units can reveal a formula error.

Before calculating, check whether the stated angle identifies the minor or major sector. For a major sector, use the reflex central angle, or subtract the minor result from the whole circle. For example, if the minor angle is $1.2$ radians, the major angle is $2\pi-1.2$ radians. This distinction changes both the arc length and the sector area proportionally while the radius remains fixed.

A common misconception is to use the degree formulas $\frac{\theta}{360}(2\pi r)$ and $\frac{\theta}{360}(\pi r^2)$ when is already in radians. Never divide a radian angle by .

IB exam technique

For a Calculate question, write the formula and substitution before the GDC value. Keep full precision until the final answer, include units, and remember that the perimeter of a sector is $r\theta+2r$, not just the arc length. All Math AI papers are GDC-active.