Illustration of circle with parts drawn to show area.

Area of a Circle by parts

Illustration of circle with parts drawn to show area.

Illustration of a circle with squares drawn to demonstrate area.

Area of Circle

Illustration of a circle with squares drawn to demonstrate area.

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height of the point P above the surface; it is required to find the distance from the point P to the horizon at A.

Circle With Center o and Radius r with point P

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height…

Illustration used to find the center of a circle.

Center of a Circle

Illustration used to find the center of a circle.

Illustration of a circle with central angle AOB.

Central Angle in a Circle

Illustration of a circle with central angle AOB.

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc."

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc." Angle AOB and angle COE are commensurable.

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc." Angle AOB and angle COE are incommensurable.

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle showing an angle included by a tangent and a chord drawn from the point of contact. The angle is measured by half the intercepted arc.

Circle With Chord and Tangent

Illustration of a circle showing an angle included by a tangent and a chord drawn from the point of…

Illustrations of a circle with a chord and triangle inside.

Circle with Chord and Triangle

Illustrations of a circle with a chord and triangle inside.

Illustration of a circle with the chord eb drawn and labeled.

Chord of Circle

Illustration of a circle with the chord eb drawn and labeled.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle AOB, and the central angle AOB has the same measure as the arc AnB.

Circle With a Chord of 2 ft. and a Radius of 3 ft.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle…

Circle with intersecting chords.

Circle With Intersecting Chords

Circle with intersecting chords.

Circle with chords.

Circle With Chords

Circle with chords.

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Chords, Diameters, and Radii of a Circle

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Chords, Secants, Diameters, and Tangents of a Circle

Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Illustration of a circle which illustrates that through three points not in a straight line one circumference, and only one, can be drawn.

Circle With Circumference Through Three Points

Illustration of a circle which illustrates that through three points not in a straight line one circumference,…

Illustrations of a circle with a circumscribed polygon. Or, a circle inscribed in an octagon.

Circle with Circumscribed Polygon

Illustrations of a circle with a circumscribed polygon. Or, a circle inscribed in an octagon.

Illustration showing the definition of a circle as a conic section. The section of a right circular cone made by a plane that is perpendicular to the axis (or parallel to the base).

Conic Section Showing A Circle

Illustration showing the definition of a circle as a conic section. The section of a right circular…

An illustration showing the intersection of a plane and a cone. The cone is intersected by a plane parallel to the base, forming a circle.

Conic Section Showing A Circle

An illustration showing the intersection of a plane and a cone. The cone is intersected by a plane parallel…

Illustration of a decagon circumscribed about a circle. This can also be described as a circle inscribed in a decagon.

Decagon Circumscribed About A Circle

Illustration of a decagon circumscribed about a circle. This can also be described as a circle inscribed…

Illustration of a decagon inscribed in a circle. This can also be described as a circle circumscribed about a decagon.

Decagon Inscribed In A Circle

Illustration of a decagon inscribed in a circle. This can also be described as a circle circumscribed…

Illustration of a circle with diameter and chord drawn. A diameter perpendicular to a chord bisects the chord and the arcs subtended by it.

Circle With Diameter and Chord

Illustration of a circle with diameter and chord drawn. A diameter perpendicular to a chord bisects…

Illustration used to show that "The diameter perpendicular to a chord bisects the chord and also its subtended arc."

Diameter Perpendicular to a Chord in a Circle

Illustration used to show that "The diameter perpendicular to a chord bisects the chord and also its…

Illustration showing a circle with a diameter, radius, and segment drawn.

Circle With Diameter, Radius, Segment

Illustration showing a circle with a diameter, radius, and segment drawn.

Illustration showing a circle with a diameter, radius, lines, triangle, and segment drawn.

Circle With Diameter, Radius, Segment, Line

Illustration showing a circle with a diameter, radius, lines, triangle, and segment drawn.

Illustration of a circle with two intersecting diameters.

Diameters of a Circle

Illustration of a circle with two intersecting diameters.

A circle divided evenly by a dotted line.

Divided Circle

A circle divided evenly by a dotted line.

Illustration of a dodecagon circumscribed about a circle. This can also be described as a circle inscribed in a dodecagon.

Dodecagon Circumscribed About A Circle

Illustration of a dodecagon circumscribed about a circle. This can also be described as a circle inscribed…

Illustration of a dodecagon inscribed in a circle. This can also be described as a circle circumscribed about a dodecagon.

Dodecagon Inscribed In A Circle

Illustration of a dodecagon inscribed in a circle. This can also be described as a circle circumscribed…

Illustration of a circle with equal chords, which are equally distant from the center.

Circle With Equal Chords

Illustration of a circle with equal chords, which are equally distant from the center.

An illustration showing how to find the center of a circle which will pass through three given points A, B, and C. "With B as a center, draw the arc DEFG; and with the same radius and A as a center, draw the cross arcs D and F; also with C as a center, draw the cross arcs E and G. Join D and F, and also E and G, and the crossing o is the required center of the circle."

Find The Center Of A Circle Through 3 Points

An illustration showing how to find the center of a circle which will pass through three given points…

Illustration of a hexagon in a circle. Four of the six vertices of the hexagon are bound by the circle (are tangent to the circle). Because all six vertices are not on the circle, the hexagon is not cyclic; it is not inscribed in the circle.

Hexagon In A Circle

Illustration of a hexagon in a circle. Four of the six vertices of the hexagon are bound by the circle…

Illustration of hexagon inscribed in circle. Or, circle circumscribed about hexagon.

Hexagon Inscribed in Circle

Illustration of hexagon inscribed in circle. Or, circle circumscribed about hexagon.

Illustration of a circle with an angle inscribed in a segment greater than a semicircle, an acute angle.

Circle With Inscribed Angle Greater Than Semicircle

Illustration of a circle with an angle inscribed in a segment greater than a semicircle, an acute angle.

Illustration of a circle with an inscribed angle.

Inscribed Angle in a Circle

Illustration of a circle with an inscribed angle.

Illustration of a circle with an angle inscribed in a segment less than a semicircle, an obtuse angle.

Circle With Inscribed Angle Less Than Semicircle

Illustration of a circle with an angle inscribed in a segment less than a semicircle, an obtuse angle.

Illustration showing the diameter of a circle inscribed in a right triangle is equal to the difference between the sum of the legs and the hypotenuse.

Circle Inscribed in a Right Triangle

Illustration showing the diameter of a circle inscribed in a right triangle is equal to the difference…

Illustration of a circle with a right angle inscribed in a semicircle.

Circle With Inscribed Right Angle in Semicircle

Illustration of a circle with a right angle inscribed in a semicircle.

Illustration showing a circle with equilateral triangles and another circle within.

Circle With Triangles and Circle Within

Illustration showing a circle with equilateral triangles and another circle within.

"A circle may be conceived as a polygon of an infinite number of sides. Thus to draw the involute of a circle divide it into a convenient number of parts, draw tangents at these points, lay off on these tangents the rectified lengths of the arch from the point of tangency to the starting point, and connect the points by a smooth curve." —French, 1911

Involute of Circle

"A circle may be conceived as a polygon of an infinite number of sides. Thus to draw the involute of…

Illustration of a regular nonagon inscribed in a circle. This could also be described as a circle circumscribed about a regular nonagon.

Nonagon Inscribed in a Circle

Illustration of a regular nonagon inscribed in a circle. This could also be described as a circle circumscribed…

Circle with diameter, radius, arc, chord, and arc.

Circle With Parts

Circle with diameter, radius, arc, chord, and arc.

Illustration of circle with arc, chord, diameter  and radius.

Parts of Circle

Illustration of circle with arc, chord, diameter and radius.

Illustration of pentagon inscribed in circle. Or, circle circumscribed about pentagon.

Pentagon Inscribed in Circle

Illustration of pentagon inscribed in circle. Or, circle circumscribed about pentagon.

Illustration of circle with perpendicular diameters.

Circle With Perpendicular Diameters

Illustration of circle with perpendicular diameters.

Illustration of circle with perpendicular diameters.

Circle With Perpendicular Diameters

Illustration of circle with perpendicular diameters.

"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter, this perpendicular will be a mean proportional between the two parts into which it divides the diameter."

Circle With a Perpendicular Drawn to the Diameter

"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter,…

Illustration of polygon circumscribed about circle. Or, circle inscribed in the polygon.

Polygon Circumscribed About Circle

Illustration of polygon circumscribed about circle. Or, circle inscribed in the polygon.

Illustration of polygon inscribed in circle. Or, circle circumscribed about the polygon.

Polygon Inscribed in Circle

Illustration of polygon inscribed in circle. Or, circle circumscribed about the polygon.

Illustration of circle with quadrant and 60 degree arc.

Quadrant of Circle With 60 degree Arc

Illustration of circle with quadrant and 60 degree arc.

Illustration of a quadrilateral circumscribed about a circle. This could also be described as a circle inscribed in a quadrilateral.

Quadrilateral Circumscribed About a Circle

Illustration of a quadrilateral circumscribed about a circle. This could also be described as a circle…

Illustration showing a circle with a quadrilateral inscribed.

Circle With Inscribed Quadrilateral

Illustration showing a circle with a quadrilateral inscribed.

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

Quadrilateral Inscribed In A Circle

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

Illustration of circle with radius of 4000, and triangle with 45 degree angle enclosed.

Circles With Radius 4000 and Enclosed Triangle

Illustration of circle with radius of 4000, and triangle with 45 degree angle enclosed.

Illustration of a circle with the radius OA drawn and labeled.

Radius of Circle

Illustration of a circle with the radius OA drawn and labeled.

Illustration of a circle with radius QR.

Radius of a Circle

Illustration of a circle with radius QR.

An illustration showing a circle with radius r, diameter d, and chord c.

Radius, Diameter, and Chord In A Circle

An illustration showing a circle with radius r, diameter d, and chord c.

Illustration of a regular heptagon/septagon inscribed in a circle. This can also be described as a circle circumscribed about a regular heptagon/septagon.

Regular Heptagon/Septagon Inscribed In A Circle

Illustration of a regular heptagon/septagon inscribed in a circle. This can also be described as a circle…

Illustration of a regular hexagon circumscribed about a circle. This can also be described as a circle inscribed in a regular hexagon.

Regular Hexagon Circumscribed About A Circle

Illustration of a regular hexagon circumscribed about a circle. This can also be described as a circle…