Diameter
The diameter of a circle is a line segment that passes through the center of the circle and has endpoints on the circle.
![](/img/a/geometry/circles/diameter/diameter.png)
Line segment AB above is a diameter for circle O since endpoints A and B lie on the circle and line segment AB passes through center O.
A circle contains infinitely many diameters. Three diameters are drawn for circle O below. As long as the line segment passes through the center and has its endpoints on the circle, it is a diameter.
![](/img/a/geometry/circles/diameter/diameters.png)
Diameter and radii
The diameter of a circle is twice the length of the circle's radius.
![](/img/a/geometry/circles/diameter/circle-diameter.png)
The two radii of length r, shown above for the circle, form diameter d. Since each radius of a circle has equal measure, d = 2r or .
Diameter and chords
A chord is a line segment whose endpoints are on a circle. If a chord contains the center, it is also a diameter of the circle. A diameter is longer than any chord that does not contain the center (i.e. is not a diameter).
![](/img/a/geometry/circles/diameter/diameter-chord.png)
Diameter AB > chord CD for circle O above since chord CD does not contain center O.
Diameter and circumference
The circumference of a circle can be compared with its diameter.
![](/img/a/geometry/circles/diameter/straight-length-of-circumference.png)
If the circumference of a circle is straightened to make a line segment shown above, its length is a little longer than the length of 3 of the circle's diameters. The circumference of a circle is its diameter scaled by pi (written symbolically as π). This is true for any circle of any size. Pi is a mathematical constant and an irrational number, so it has an infinite number of digits, and no repeating pattern. It is approximately equal to 3.14159.
Diameter and semi-circles
An arc of a circle whose endpoints lie on the endpoints of a diameter of the circle forms a semi-circle or half-circle.
![](/img/a/geometry/circles/diameter/semicircle.png)
The endpoints of arc ABC for circle O shown above lie on the endpoints of diameter AB, forming a semi-circle. Additionally, a diameter divides a circle into two semi-circles, so the arc below arc ABC is also a semi-circle.
Since the semi-circle encompasses one-half of the entire circle, its circumference is one-half the circumference of the circle, or πr, where r is the radius of the circle.
Other properties and theorems
If the vertex of a right angle lies on a circle, then the points of intersection of the sides of the angle and the circle form a diameter of the circle.
![](/img/a/geometry/circles/diameter/right-angle.png)
The vertex of right angle B is on circle O, shown above. The measure of an inscribed angle is one-half the measure of its intercepted arc. Since angle B is an inscribed angle and its measure is 90°, intercepted arc has a measure of 180°. Thus, the arc is a semicircle and chord AC is a diameter of circle O.
Conversely, if the vertices of a triangle are on a circle and one side of the triangle is a diameter of the circle, then the angle opposite of the diameter is a right angle and the triangle is a right triangle.
If a chord of a circle is the perpendicular bisector of another chord, the chord that is the perpendicular bisector is a diameter of the circle.
![](/img/a/geometry/circles/diameter/perpendicular-bisector.png)
Chord CD is the perpendicular bisector of chord AB for circle O, shown above, so chord CD is a diameter of the circle.
Diameter of spheres
A chord for a sphere is a line segment whose endpoints are on the sphere. If the chord contains the center of the sphere, it is a diameter of the sphere.
![](/img/a/geometry/circles/diameter/sphere-diameter.png)
Two chords, chord AB and chord CD, are drawn inside the sphere with center O, shown above. Since chord AB contains the center of the sphere, chord AB is a diameter of the sphere. A diameter of a sphere lies on a great circle of the sphere. A great circle divides a sphere into two semi-spheres (half spheres).
![](/img/a/geometry/circles/diameter/semisphere.png)
The semi-sphere above the great circle containing diameter AB is shown above. The semi-spheres above and below the great circle are the same size. This is true for any two semi-spheres of a sphere.