🔵 Circles: Area & Circumference
Complete SAT Math Guide — Every Formula, Strategy & Problem Type
SAT Math
Geometry
High-Yield Topic
Part 1 — What Is a Circle?
Definition
A circle is the set of all points in a plane that are the same distance from a fixed point called the center.
| Term | Symbol | Meaning |
|---|---|---|
| Center | $(h,k)$ | The fixed point; every point on the circle is equidistant from it |
| Radius | $r$ | Distance from center to any point on the circle |
| Diameter | $d$ | Distance across the circle through the center — the longest chord |
| Circumference | $C$ | Distance around the circle (its perimeter) |
| Area | $A$ | Measure of the region enclosed by the circle |
Key Relationship: $d = 2r$ ⟹ $r = \dfrac{d}{2}$
What Is π (Pi)?
$\pi$ is the ratio of any circle's circumference to its diameter:
$$\pi = \frac{C}{d} \approx 3.14159\ldots$$It is an irrational number — its decimal never terminates or repeats.
- Common approximations: $\pi \approx 3.14$ or $\pi \approx \frac{22}{7}$
- SAT Tip: Most answers are left in terms of $\pi$ (e.g., $25\pi$).
Interactive Circle Diagram
Interactive visualization — available in the full lesson
Green = Radius
Orange = Diameter
Purple = Circumference
Part 2 — Circumference (Perimeter of a Circle)
$$\boxed{C = 2\pi r = \pi d}$$
Since $d = 2r$, both formulas are equivalent: $C = \pi d = \pi(2r) = 2\pi r$.
Circumference is linear — measured in length units (cm, m, ft, etc.).
Quick Examples
| Given | Work | Exact $C$ | Approx $C$ |
|---|---|---|---|
| $r = 5$ | $2\pi(5)$ | $10\pi$ | $\approx 31.42$ |
| $r = 7$ | $2\pi(7)$ | $14\pi$ | $\approx 43.98$ |
| $r = 3$ | $2\pi(3)$ | $6\pi$ | $\approx 18.85$ |
| $d = 10$ | $\pi(10)$ | $10\pi$ | $\approx 31.42$ |
| $d = 14$ | $\pi(14)$ | $14\pi$ | $\approx 43.98$ |
| $d = 8$ | $\pi(8)$ | $8\pi$ | $\approx 25.13$ |
Finding Radius or Diameter FROM Circumference
$$r = \frac{C}{2\pi} \qquad d = \frac{C}{\pi}$$
| Given $C$ | Find | Work | Answer |
|---|---|---|---|
| $20\pi$ | $r$ | $\frac{20\pi}{2\pi}$ | $r = 10$ |
| $18\pi$ | $d$ | $\frac{18\pi}{\pi}$ | $d = 18$ |
| $12\pi$ | $r$ | $\frac{12\pi}{2\pi}$ | $r = 6$ |
| $50$ | $r$ | $\frac{50}{2\pi}$ | $r = \frac{25}{\pi} \approx 7.96$ |
| $31.4$ | $d$ | $\frac{31.4}{3.14}$ | $d \approx 10$ |
⚠️ SAT Trap: If the problem gives you the diameter, don't plug it into $C = 2\pi r$ without dividing by 2 first! Always identify whether you have $r$ or $d$.