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An illustration depicting an infinite sequence of tangent circles with the radius converging to zero. This is often called a Hawaiian earring.

Infinite Tangent Circles

An illustration depicting an infinite sequence of tangent circles with the radius converging to zero.…

Illustration of a 6-point star (convex dodecagon) inscribed in a large circle and circumscribed about a smaller circle.

Star Inscribed And Circumscribed About Circles

Illustration of a 6-point star (convex dodecagon) inscribed in a large circle and circumscribed about…

Illustration of an 8-point star (convex polygon) inscribed in a large circle and circumscribed about a smaller circle.

Star Inscribed And Circumscribed About Circles

Illustration of an 8-point star (convex polygon) inscribed in a large circle and circumscribed about…

Illustration of a cone of revolution used to show that the lateral area is equal to half the product of the slant height by the circumference of the base.

Lateral Area of Cone of Revolution

Illustration of a cone of revolution used to show that the lateral area is equal to half the product…

Illustration of a pyramid circumscribed about a cone.

Pyramid Circumscribed About a Cone

Illustration of a pyramid circumscribed about a cone.

Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If a pyramid whose base is a regular polygon is inscribed in or circumscribed about a circular cone, and if the number of sides of the base of the pyramid is indefinitely increased, the volume of the cone is the limit of the volume of the pyramid, and the lateral area of the cone is the limit of the lateral area of the pyramid."

Cone With Regular Polygon Inscribed and Circumscribed About

Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If…

Illustration of a pyramid inscribed in a cone.

Pyramid Inscribed in a Cone

Illustration of a pyramid inscribed in a cone.

Illustration of a cone with a polygon inscribed used to show that the volume of a circular cone is equal to one third the product of its base by its altitude.

Volume of Cone

Illustration of a cone with a polygon inscribed used to show that the volume of a circular cone is equal…

Right circular cylinder inscribed in a pentagonal prism. Or, Pentagonal prism circumscribed about a cylinder.

Cylinder Inscribed in Pentagonal Prism

Right circular cylinder inscribed in a pentagonal prism. Or, Pentagonal prism circumscribed about a…

Illustration of a prism circumscribed about a cylinder.

Prism Circumscribed About A Cylinder

Illustration of a prism circumscribed about a cylinder.

Illustration of a triangular prism inscribed in a cylinder.

Prism inscribed in Cylinder

Illustration of a triangular prism inscribed in a cylinder.

Illustration of a triangular prism inscribed in a cylinder.

Prism inscribed in Cylinder

Illustration of a triangular prism inscribed in a cylinder.

Illustration of a pentagonal prism inscribed in a cylinder.

Prism inscribed in Cylinder

Illustration of a pentagonal prism inscribed in a cylinder.

A large cylinder containing 2 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

2 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 2 smaller congruent cylinders. The small cylinders are externally tangent…

A large cylinder containing 3 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

3 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 3 smaller congruent cylinders. The small cylinders are externally tangent…

A large cylinder containing 4 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

4 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 4 smaller congruent cylinders. The small cylinders are externally tangent…

A large cylinder containing 7 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

7 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 7 smaller congruent cylinders. The small cylinders are externally tangent…

Design made by drawing one large circle and then two circles that are vertically placed and internally tangent to the original circle. Erase the left side of the top circle and the right side of the bottom circle to create the design. It resembles the yin and yang symbol.

Design Similar to Yin Yang Symbol

Design made by drawing one large circle and then two circles that are vertically placed and internally…

Circular design made by rotating circles about a fixed point. The radii of the smaller circles is equal to the distance between the point of rotation and the center of the circle. The circles meet in the center of the larger circle. The design is achieved by removing consecutive halves of the circles (semi-circles).

Circular Design

Circular design made by rotating circles about a fixed point. The radii of the smaller circles is equal…

Circular design made by rotating circles about a fixed point. The radii of the smaller circles is equal to the distance between the point of rotation and the center of the circle. The circles meet in the center of the larger circle. The design is achieved by removing consecutive halves of the circles (semi-circles).

Circular Design

Circular design made by rotating circles about a fixed point. The radii of the smaller circles is equal…

Illustration of 12 equilateral triangles inscribed in an equilateral dodecagon. Each vertex of the dodecagon contains a vertex of a triangle.

12 Triangles Inscribed In A Dodecagon

Illustration of 12 equilateral triangles inscribed in an equilateral dodecagon. Each vertex of the dodecagon…

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be described as a regular dodecagon circumscribed about a 12-point star (24-sided polygon).

12-Point Star Inscribed In A Dodecagon

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be…

Illustration of 3 squares inscribed in an regular dodecagon. Each vertex of the dodecagon is also a vertex of one of the squares.

3 Square Inscribed In A Dodecagon

Illustration of 3 squares inscribed in an regular dodecagon. Each vertex of the dodecagon is also a…

Illustration of a regular hexagon and an equilateral triangle inscribed in a regular dodecagon. This could also be described as an equilateral triangle inscribed in a regular hexagon, which is inscribed in a regular dodecagon.

Hexagon And Triangle Inscribed In A Dodecagon

Illustration of a regular hexagon and an equilateral triangle inscribed in a regular dodecagon. This…

Illustration of a regular hexagon inscribed in a regular dodecagon. This could also be described as a regular dodecagon circumscribed about a regular hexagon.

Hexagon Inscribed In A Dodecagon

Illustration of a regular hexagon inscribed in a regular dodecagon. This could also be described as…

Illustration of a square circumscribed about a regular dodecagon. This could also be described as a dodecagon inscribed in a square.

Square Circumscribed About A Dodecagon

Illustration of a square circumscribed about a regular dodecagon. This could also be described as a…

Illustration of a square inscribed in an regular dodecagon. This could also be described as a regular dodecagon circumscribed about a square.

Square Inscribed In A Dodecagon

Illustration of a square inscribed in an regular dodecagon. This could also be described as a regular…

Illustration of 2 squares; one inscribed in a regular dodecagon and the other circumscribed about the same dodecagon.

Squares Inscribed and Circumscribed About a Regular Dodecagon

Illustration of 2 squares; one inscribed in a regular dodecagon and the other circumscribed about the…

Illustration of an equilateral triangle inscribed in an equilateral dodecagon. This could also be described as a dodecagon circumscribed about an equilateral triangle.

Triangle Inscribed In A Dodecagon

Illustration of an equilateral triangle inscribed in an equilateral dodecagon. This could also be described…

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal to the length of the major axis of the ellipse. The ellipse is externally tangent to the circle.

Ellipse Inscribed In A Circle

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal…

Illustration of a rhombus circumscribed about an ellipse. This could also be described as an ellipse inscribed in a rhombus.

Rhombus Circumscribed About an Ellipse

Illustration of a rhombus circumscribed about an ellipse. This could also be described as an ellipse…

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments, are externally tangent to the circle. The illustration could be used as a 3-dimensional drawing of a sphere.

2 Ellipses Inscribed In A Circle

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been rotated 12° in relation to the one next to it. The outer vertices are connected with a smoother curve to form a circle. Hence, the circle is circumscribed about the triangles.

10 Congruent Rotated Equilateral Triangles

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been…

Illustration of beginning of construction of a regular heptagon in a circle.

Construction of Regular Heptagon in a Circle

Illustration of beginning of construction of a regular heptagon in a circle.

Illustration used to show how to inscribe a regular hexagon in a given circle.

Construction Of Hexagon Inscribed In Circle

Illustration used to show how to inscribe a regular hexagon in a given circle.

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

Illustration of regular hexagon and triangle inscribed in circle.

Hexagon Inscribed in Circle by Construction

Illustration of regular hexagon and triangle inscribed in circle.

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

Circle Inscribed in a Nonagon

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

Regular nonagon inscribed in a circle.

Nonagon Inscribed in a Circle

Regular nonagon inscribed in a circle.

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

Triangle Inscribed in a Nonagon

Illustration of an equilateral triangle inscribed in a regular nonagon. This could also be described…

Illustration used to show how to inscribe a regular octagon in a given circle.

Construction Of Octagon Inscribed In Circle

Illustration used to show how to inscribe a regular octagon in a given circle.

Illustration of a regular octagon circumscribed about a square. This could also be described as a square inscribed in a regular octagon.

Octagon Circumscribed About a Square

Illustration of a regular octagon circumscribed about a square. This could also be described as a square…

Illustration used to show how to inscribe a regular pentagon in a given circle.

Construction Of Pentagon Inscribed In Circle

Illustration used to show how to inscribe a regular pentagon in a given circle.

Regular pentagon inscribed in a circle, circumscribed about a circle.

Pentagon Inscribed & Circumscribed in Circles

Regular pentagon inscribed in a circle, circumscribed about a circle.

Illustration of a cyclic pentagon, a pentagon inscribed in a circle. This can also be described as a circle circumscribed about a pentagon. In this illustration, the pentagon is not regular (the lengths of the sides are not equal).

Cyclic Pentagon

Illustration of a cyclic pentagon, a pentagon inscribed in a circle. This can also be described as a…

Illustration showing how the golden ratio in a regular pentagon (inscribed in a circle) can be found using Ptolemy's theorem. The lines that are bolded form a quadrilateral. Ptolemy's theorem says the square of b equals the sum of a squared and ab, which in turn gives the golden ratio.

Golden Ratio In A Pentagon

Illustration showing how the golden ratio in a regular pentagon (inscribed in a circle) can be found…

Illustration of pentagon inscribed in circle. Or, circle circumscribed about pentagon. Construction lines shown.

Pentagon Inscribed in Circle by Construction

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

Illustration showing a circle inscribed in a regular pentagon. Or, a regular pentagon circumscribed about a circle.

Regular Pentagon With Circle Inscribed

Illustration showing a circle inscribed in a regular pentagon. Or, a regular pentagon circumscribed…

Illustration used to show how to inscribe any regular polygon in a given circle.

Construction Of Polygon Inscribed In Circle

Illustration used to show how to inscribe any regular polygon in a given circle.

Illustration of a cyclic quadrilateral, a quadrilateral inscribed in a circle. This can also be described as a circle circumscribed about a quadrilateral. In this illustration, the quadrilateral is not regular (the lengths of the sides are not equal).

Cyclic Quadrilateral

Illustration of a cyclic quadrilateral, a quadrilateral inscribed in a circle. This can also be described…

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

Regular Heptagon/Septagon Circumscribed about a Circle

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

Circular rosette with 12 petals in a circle. It is made by rotating circles about a fixed point. The radii of the smaller circles are less than the distance between the point of rotation and the center of the circle. Thus, there is a hole in the center.

Circular Rosette With 12 Petals

Circular rosette with 12 petals in a circle. It is made by rotating circles about a fixed point. The…

Circular rosette with 12 petals. It is made by rotating circles about a fixed point. The radii of the smaller circles is equal to the distance between the point of rotation and the center of the circle. Thus, the circles meet in the center.

Circular Rosette With 12 Petals

Circular rosette with 12 petals. It is made by rotating circles about a fixed point. The radii of the…

Circular rosette with 16 petals in a circle. It is made by rotating circles about a fixed point. The radii of the smaller circles is equal to the distance between the point of rotation and the center of the circle. Thus, the circles meet in the center of the larger circle.

Circular Rosette With 16 Petals

Circular rosette with 16 petals in a circle. It is made by rotating circles about a fixed point. The…

Circular rosette with 16 petals in a circle. It is made by rotating circles about a fixed point. The radii of the smaller circles are less than the distance between the point of rotation and the center of the circle. Thus, there is a hole in the center.

Circular Rosette With 16 Petals

Circular rosette with 16 petals in a circle. It is made by rotating circles about a fixed point. The…

Circular rosette with 24 petals in a circle. It is made by rotating circles about a fixed point. The radii of the smaller circles are less than the distance between the point of rotation and the center of the circle. Thus, there is a hole in the center.

Circular Rosette With 24 Petals

Circular rosette with 24 petals in a circle. It is made by rotating circles about a fixed point. The…

Circular rosette with 24 petals. It is made by rotating circles about a fixed point. The radii of the smaller circles is equal to the distance between the point of rotation and the center of the circle. Thus, the circles meet in the center.

Circular Rosette With 24 Petals

Circular rosette with 24 petals. It is made by rotating circles about a fixed point. The radii of the…