Table of Contents
Basic idea of a circle
A circle is the set of all points in a plane that are the same distance from a fixed point.
- The fixed point is called the center.
- The fixed distance is called the radius.
If the center is point $O$ and $r$ is a positive number, then the circle with center $O$ and radius $r$ is the set of all points $P$ such that the distance $OP$ equals $r$.
We usually name a circle by its center, for example “circle $O$.”
Key parts of a circle
A circle has several important parts. Some will be used again and again in later geometry.
- Center: The fixed point in the middle of the circle. All points on the circle are the same distance from the center.
- Radius: A line segment from the center to a point on the circle.
- Plural: radii (pronounced “ray-dee-eye”).
- All radii of the same circle have the same length.
- The length of a radius is usually written as $r$.
- Diameter: A line segment that passes through the center and has its endpoints on the circle.
- The length of a diameter is usually written as $d$.
- Every diameter is made of two radii laid end to end.
- Relationship:
$$
d = 2r
\quad\text{and}\quad
r = \frac{d}{2}.
$$ - Chord: A line segment with both endpoints on the circle.
- Every diameter is a chord, but not every chord is a diameter.
- Among all chords in a circle, the diameter is the longest.
- Arc: A connected part of the circle itself (a “curved segment” of the circumference).
- A minor arc is less than half the circle.
- A major arc is more than half the circle.
- A semicircle is exactly half the circle; its endpoints lie at the ends of a diameter.
- Circumference: The full distance around the circle (the “perimeter” of the circle).
- The circumference is the whole curve of the circle.
- Its length is usually written as $C$.
- Secant: A line that intersects the circle at two points. (Unlike a chord, a secant is an infinite line, not just the segment between the intersection points.)
- Tangent: A line that touches the circle at exactly one point.
- The single point where it touches is the point of tangency.
Radius, diameter, and circumference
The most important relationships in a circle link radius, diameter, and circumference.
Using $r$ for radius, $d$ for diameter, $C$ for circumference:
- Diameter and radius:
$$
d = 2r, \quad r = \frac{d}{2}.
$$ - Circumference and diameter:
$$
\frac{C}{d} = \pi,
$$
so
$$
C = \pi d.
$$ - Combining with $d = 2r$, we also get:
$$
C = 2\pi r.
$$
Here $\pi$ (pi) is a constant. Its exact value cannot be written as a simple fraction, but its decimal approximation begins $3.14159\ldots$ and goes on forever without repeating.
When working with beginners’ problems, it is common to:
- either leave answers in terms of $\pi$, such as $C = 8\pi$,
- or approximate $\pi \approx 3.14$ (unless a different instruction is given).
Naming and notation for circles
Common ways to refer to a circle and its parts include:
- Circle name: If $O$ is the center, we write “circle $O$,” sometimes written as $\bigcirc O$ in text.
- Radius: If $O$ is the center and $A$ is a point on the circle, $OA$ is a radius, and its length is written $OA$ or $|OA|$.
- Diameter: If $A$ and $B$ are opposite points on the circle and $O$ is the center, then $AB$ is a diameter.
- Arc: An arc from $A$ to $B$ on the circle can be written as $\widehat{AB}$.
- Chord: A chord from $A$ to $B$ is the segment $AB$ with both endpoints on the circle.
Circles in coordinate geometry
You will see circles again in analytic geometry, where points have coordinates. The simplest standard form of a circle with center $(h,k)$ and radius $r$ is the set of all points $(x,y)$ satisfying
$$
(x - h)^2 + (y - k)^2 = r^2.
$$
For a circle centered at the origin $(0,0)$, this simplifies to
$$
x^2 + y^2 = r^2.
$$
Understanding this equation will be important later, but at this stage you mainly need to recognize that it describes a circle and that $r$ is its radius.
Basic properties and symmetry
Although deeper properties will be treated in later circle-related sections, some simple geometric facts are useful to know early:
- All radii in a circle are congruent (have equal length).
- Any two diameters of the same circle are congruent.
- A circle is symmetric in many ways:
- It looks the same after rotation about its center by any angle.
- Reflecting it across any line that passes through the center leaves it unchanged.
- The center of a circle is the unique point that is the same distance from all points on the circle; this helps explain why constructions often focus on finding or using the center.
These basic ideas form the foundation for more specific topics such as “Radius and diameter” and “Arcs and sectors,” which will be covered separately.