In Circumference and Pi you measured around circles. Here we name their parts carefully, using the language of points, lines and segments from Points, Lines and Planes. A circle is the set of all points in a plane at one fixed distance, the radius, from one point, the center. A circle with center $O$ is called circle $O$, written $\odot O$.
| Part | Definition |
|---|---|
| radius | a segment from the center to a point on the circle (also its length, $r$) |
| chord | a segment whose endpoints are both on the circle |
| diameter | a chord through the center; its length is $d = 2r$ |
| secant | a line that crosses the circle at two points |
| tangent | a line in the plane of the circle that touches it at exactly one point, the point of tangency |
| arc | part of the circle between two points; a minor arc is less than half the circle, a major arc more than half, and a semicircle exactly half |
| central angle | an angle with its vertex at the center |
Por qué funciona
Every part is defined from the center and the radius, because that is what a circle is: all its points are one radius from the center. That single fact will explain every theorem in this unit, from why all radii are equal to why a tangent meets a radius at a right angle.
Idea clave
Radii go from the center to the circle; chords join two points of the circle; a diameter is the chord through the center ($d = 2r$, and it is the longest chord). A secant line cuts the circle twice; a tangent line touches it once.
Ejemplo resuelto: Naming parts
In $\odot O$, $\overline{PQ}$ has both endpoints on the circle and passes through $O$. Line $\ell$ meets the circle only at $T$. What are they?
- A segment with both ends on the circle is a chord; one through the center is a diameter. So $\overline{PQ}$ is a diameter.
- A line meeting the circle at exactly one point is a tangent, with point of tangency $T$.
Respuesta: $\overline{PQ}$ is a diameter; $\ell$ is a tangent.
Ejemplo resuelto: Radius and diameter
A circular table has a diameter of $54$ in. What is its radius? If a different circle has circumference $18\pi$ cm, what is its radius?
- $r = \frac{d}{2} = 27$ in.
- $C = 2\pi r = 18\pi$, so $r = 9$ cm.
Respuesta: $27$ in and $9$ cm
Error común
Calling every chord a diameter, or calling a whole line a chord. A chord is a segment; only the chord through the center is a diameter. A line that extends past the circle is a secant. Check: does it pass through the center? Does it stop at the circle?