Math Core

Wheels, clock faces, pizza, the orbit of a satellite: circles show up everywhere. Before you can find angles, lengths and areas in circles, you need a shared vocabulary for the segments and lines that belong to them. This lesson builds that vocabulary and connects the radius to the distance around a circle.

What a circle is

You probably picture a circle as a round shape, but geometry needs a more precise description. Think about how you would draw one with a string and a pencil: pin one end of the string down, pull it tight, and swing the pencil all the way around. Every point you draw is the same distance from the pin.

Definition

Circle

A circle is the set of all points in a plane that are the same distance from a fixed point, called the center. A circle is named by its center: a circle with center $O$ is called $\odot O$ ("circle $O$ ").

Notice that the circle is only the curve itself. The center is not on the circle, and neither are the points inside it. A point is inside a circle if its distance from the center is less than the radius, on the circle if the distance equals the radius, and outside if the distance is greater.

Segments and lines in a circle

The figure shows $\odot O$ with several important segments and lines.

Circle O with radius OA, diameter BC, chord DE and a line tangent to the circle at T.

term	what it is	in the figure
radius	a segment from the center to a point on the circle	$\overline{OA}$
chord	a segment whose endpoints are both on the circle	$\overline{DE}$
diameter	a chord that passes through the center	$\overline{BC}$
secant	a line that crosses the circle at two points	line $BC$
tangent	a line in the plane of the circle that touches it at exactly one point	the line through $T$
point of tangency	the one point where a tangent touches the circle	$T$

The words radius and diameter also name lengths. "The radius is $5$ " means every radius segment has length $5$ . Since a diameter is made of two radii placed end to end through the center,

$d = 2r \qquad \text{and} \qquad r = \dfrac{d}{2}.$

A diameter is the longest possible chord of a circle. Any other chord, like $\overline{DE}$ , is shorter.

Common mistake

A chord is a segment and a secant is a line. Every secant contains a chord (the part inside the circle), but the secant keeps going past the circle in both directions. Also, not every chord is a diameter: a chord has to pass through the center to be a diameter.

Relationships between circles

Two circles are congruent if they have the same radius. They are concentric if they share the same center, like the rings of a target. Concentric circles with different radii are not congruent.

Two coplanar circles can meet at two points, one point, or no points. A line that is tangent to both circles is a common tangent.

Circumference

The distance around a circle is its circumference. For every circle, the ratio of the circumference to the diameter is the same number, called $\pi$ (pi), which is about $3.14159$ .

Circumference

For a circle with radius $r$ and diameter $d$ ,

$C = \pi d = 2\pi r.$

In geometry you will often leave answers in terms of $\pi$ , such as $10\pi$ , because that is exact. Use a decimal only when a problem asks you to round.

Worked example: Radius, diameter and circumference

A circle has radius $6$ cm. Find its diameter and its circumference.

The diameter is twice the radius: $d = 2(6) = 12$ cm.

The circumference is $C = 2\pi r = 2\pi(6) = 12\pi$ cm, which is about $37.7$ cm.

Worked example: Working backward from the circumference

A circular track has a circumference of $400\pi$ meters. Find its radius.

Set up $2\pi r = 400\pi$ and divide both sides by $2\pi$ :

$r = \dfrac{400\pi}{2\pi} = 200 \text{ meters}.$

Worked example: An algebra connection

In $\odot P$ , a radius has length $2x + 1$ and a diameter has length $5x - 4$ . Find the radius.

A diameter is twice a radius, so

\begin{aligned} 5x - 4 &= 2(2x + 1) \\ 5x - 4 &= 4x + 2 \\ x &= 6. \end{aligned}

The radius is $2(6) + 1 = 13$ . Check: the diameter is $5(6) - 4 = 26 = 2 \cdot 13$ .

Tip

Before you compute anything, ask yourself: "Was I given the radius or the diameter?" Mixing them up doubles or halves every answer. If you're given the diameter, write $r = d/2$ first.

Practice

Practice 1

In $\odot O$ , $\overline{PQ}$ has both endpoints on the circle and passes through $O$ . What is $\overline{PQ}$ ?

Practice 2

A circle has radius $7$ inches. What is its diameter, in inches?