Arcs, Sectors, and Central Angles

When working with circles, it is important to understand the relationship between the angles at the center of the circle and the pieces of the circle they create.

Degrees and Radians

Before calculating lengths and areas, you need to know how to measure angles. Angles can be measured in degrees or radians. A full circle is $360^\circ$, which is exactly equal to $2\pi$ radians.

This gives us the fundamental conversion ratio: $180^\circ = \pi \text{ radians}$

• To convert radians to degrees: Multiply by $\frac{180^\circ}{\pi}$
• To convert degrees to radians: Multiply by $\frac{\pi}{180^\circ}$

Example: Convert $\frac{5\pi}{6}$ radians to degrees. $\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = \frac{5 \times 180^\circ}{6} = 150^\circ$

Arc Length

An arc is a portion of the circumference of a circle. A central angle is an angle whose vertex is at the center of the circle. The length of an arc ($s$) is directly proportional to its central angle ($\theta$).

If the angle is in degrees, the arc length is a fraction of the total circumference ($2\pi r$): $s = \frac{\theta}{360^\circ} \times 2\pi r$

If the angle is in radians, the formula simplifies beautifully to: $s = r\theta$

Example: Find the arc length of a $120^\circ$ arc in a circle with radius 9. $s = \frac{120^\circ}{360^\circ} \times 2\pi(9) = \frac{1}{3} \times 18\pi = 6\pi$

Area of a Sector

A sector is a "slice of pie" formed by two radii and the intercepted arc. Just like arc length, the area of a sector ($A$) is proportional to the central angle.

If the angle is in degrees, the sector area is a fraction of the total circle area ($\pi r^2$): $A = \frac{\theta}{360^\circ} \times \pi r^2$

If the angle is in radians, the formula is: $A = \frac{1}{2}r^2\theta$

By matching the angle measure (degrees or radians) to the correct formula, you can easily find the exact size of any slice of a circle.