Circles: use circumference and area, compute arc length and sector area as fractions of the whole, convert between degrees and radians, and apply central angle relationships.

What this skill is asking

Circle questions on the Digital SAT (Geometry and Trigonometry domain) cover circumference and area, arc length and sector area as fractions of the whole circle, and radian measure. The circumference and area formulas are on the reference sheet, along with the degree-radian facts; the skill is the fraction-of-the-whole reasoning and the degree-radian conversion.

The circle formulas

Two formulas plus the fraction idea handle everything.

A worked sector-area question

Sectors are a fraction of the circle's area.

Arc length, sector area, and the fraction idea

The single idea behind arcs and sectors is proportion: an arc or sector is the same fraction of the whole circle as its central angle is of the full $360^\circ$. So an arc is $\frac{\theta}{360}$ of the circumference and a sector is $\frac{\theta}{360}$ of the area. This one principle replaces two separate formulas: compute the whole (circumference or area), then multiply by the angle fraction. The reference sheet even states the relationship as $\frac{\text{arc}}{\text{circumference}} = \frac{\text{central angle}}{360^\circ}$. Setting up that proportion and solving for whichever quantity is unknown handles every arc-and-sector question.

Degrees and radians

The SAT uses both degrees and radians, and you must convert between them. The anchor fact, on the reference sheet, is that a full circle is $360^\circ = 2\pi$ radians. From it, degrees to radians multiplies by $\frac{\pi}{180}$, and radians to degrees multiplies by $\frac{180}{\pi}$. Common values are worth knowing on sight: $90^\circ = \frac{\pi}{2}$, $180^\circ = \pi$, $60^\circ = \frac{\pi}{3}$, $45^\circ = \frac{\pi}{4}$, $30^\circ = \frac{\pi}{6}$. In radians the arc-length formula simplifies to $s = r\theta$ (no $360$ fraction), which is why radian measure is the natural unit for arcs. When a question mixes the two, convert to a single unit first.

Central angles and the proportion in reverse

The arc-and-sector proportion also runs backward: given an arc length or sector area, you can solve for the central angle or the radius. If an arc of length $4\pi$ sits on a circle of radius $12$, then $\frac{\theta}{360} \times 2\pi(12) = 4\pi$, so $\frac{\theta}{360} \times 24\pi = 4\pi$, giving $\frac{\theta}{360} = \frac{1}{6}$ and $\theta = 60^\circ$. The same structure recovers the radius if the angle and arc are known. A useful related fact is that a central angle equals the measure of the arc it cuts off, so the arc "measures" $60^\circ$ in the example, which is how questions describe arcs by angle. Setting up the single proportion $\frac{\text{arc}}{\text{circumference}} = \frac{\text{central angle}}{360^\circ}$ and solving for whichever piece is unknown handles arc, sector, angle, and radius questions alike.