Circles — Core Skills

Circles: The Basics

Part 1 of 2 — One Skill, One Idea

Two words do most of the work in circle questions.

• The radius is the distance from the center of the circle out to the edge.
• The diameter is the distance all the way across, through the center.

The diameter is twice the radius. So if you know one, you know the other:

$d = 2r \qquad \text{and} \qquad r = \frac{d}{2}$

If the radius is $4$, the diameter is $8$. If the diameter is $8$, the radius is $4$.

The two formulas

• Area (the space inside): $A = \pi r^{2}$
• Circumference (the distance around the edge): $C = 2\pi r$

Both formulas use the radius, not the diameter. When a question gives you a diameter, cut it in half first. That single habit prevents most circle mistakes.

The symbol $\pi$ is a number, about $3.14$. On the SAT you can usually leave it as $\pi$ in your answer.

Worked example

A circle has radius $3$. Find its area and its circumference.

Area — put $3$ into $A = \pi r^{2}$. Square the radius first, then multiply by $\pi$:

$3^{2} = 9$

$A = 9\pi$

Circumference — put $3$ into $C = 2\pi r$:

$C = 2 \times \pi \times 3 = 6\pi$

Same circle, two different answers: $9\pi$ for area and $6\pi$ for circumference. Check which one the question wants.

One reminder about the area formula: the exponent applies to $r$ only. In $\pi r^{2}$ you square the radius, then multiply by $\pi$ — you never square $\pi$.

The Equation of a Circle

Part 2 of 2 — Practice

The formulas from Part 1

• Radius $=$ center to edge. Diameter $=$ all the way across. $d = 2r$ and $r = \frac{d}{2}$.