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 $r = \frac{d}{2}$.
• Area: $A = \pi r^{2}$
• Circumference: $C = 2\pi r$
• Both formulas use the radius. Given a diameter, halve it first.

Circles on a grid

A circle drawn on the $xy$-plane has an equation that looks like this:

$(x - h)^{2} + (y - k)^{2} = r^{2}$

You do not have to build this equation from scratch. You only have to read it. Three things to know:

• $(h, k)$ is the center.
• The number on the right is $r^{2}$ — the radius squared. Take its square root to get the radius.
• The signs flip. $(x - 3)^{2}$ means the center's $x$ is $+3$. $(x + 3)^{2}$ means the center's $x$ is $-3$.

Worked example

$(x - 2)^{2} + (y - 5)^{2} = 36$

Center — read the numbers and flip their signs: $x$ is $2$, $y$ is $5$. The center is $(2, 5)$.

Radius — the right side is $36$, and that is $r^{2}$. So:

$r = \sqrt{36} = 6$

The radius is $6$, not $36$. This is the single most common circle mistake, so it is worth pausing every time: the number on the right is squared already.

A shorter version

When the center is the origin, $(0, 0)$, both $h$ and $k$ are $0$, so the equation gets shorter:

$x^{2} + y^{2} = r^{2}$

For example, $x^{2} + y^{2} = 49$ is a circle centered at the origin with radius $\sqrt{49} = 7$.