Spheres and Spherical Geometry - Theory & Concepts

Spheres and Spherical Geometry

Learning Objectives

Understand the fundamental elements of spherical geometry, such as great circles and small circles.
Apply formulas for the volume and surface area of spheres, spherical shells, caps, and zones.
Calculate areas of spherical polygons and volumes of spherical pyramids and wedges.
Understand and apply Archimedes' relationship between a sphere, inscribed cone, and circumscribed cylinder.

Fundamental concepts and formulas for spherical solids, surfaces, and complex spherical shapes.

Great Circle

A circle formed by the intersection of a sphere with a plane passing through its center. It is the largest possible circle that can be drawn on the sphere.

Small Circle

A circle formed by the intersection of a sphere with a plane that does not pass through its center.

Properties of Spherical Geometry

Understanding spherical geometry involves recognizing specific elements of a sphere, such as great circles and small circles, which serve as the foundation for measuring distances, areas, and volumes on spherical surfaces.

Archimedes' Volumetric Relationship

A profound discovery by Archimedes links the sphere, the circumscribing cylinder (a cylinder with height and diameter equal to the sphere's diameter, $h=2r$ , $D=2r$ ), and the inscribed cone (same base and height). Their volumes have a simple integer ratio of exactly 1:2:3 for Cone:Sphere:Cylinder.

Volume of Cone = $\frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3$
Volume of Sphere = $\frac{4}{3}\pi r^3$
Volume of Cylinder = $\pi r^2 (2r) = 2\pi r^3$

This means the volume of the sphere is exactly $\frac{2}{3}$ the volume of the circumscribing cylinder.

Interactive Simulation

Use the simulation below to explore the volume ratio relationship between the cone, sphere, and cylinder described by Archimedes.

Volume of a Sphere

The volume of a perfect sphere.

V = \frac{4}{3}\pi r^3

Variables

Symbol	Description	Unit
$V$	Volume	-
$r$	Radius	-

Surface Area of a Sphere

The total outer area of a sphere.

A = 4\pi r^2

Variables

Symbol	Description	Unit
$A$	Surface area	-
$r$	Radius	-

Volume of a Spherical Shell

A spherical shell bounds a region between two concentric spheres of different radii. The material volume is the difference in volume between the outer and inner spheres.

V = \frac{4}{3}\pi (R^3 - r^3)

Variables

Symbol	Description	Unit
$V$	Volume of material	-
$R$	Outer radius	-
$r$	Inner radius	-
$t$	Thickness of the shell ( $t = R - r$ )	-

Spherical Polygon

A polygon on the surface of a sphere whose sides are arcs of great circles.

Area of a Spherical Polygon and Spherical Triangle

The area of a spherical polygon or triangle relies on the concept of spherical excess ( $E$ ). The sum of interior angles ( $s$ ) in a spherical triangle is strictly bounded: $180^\circ < s < 540^\circ$ .

E = s - (n - 2) \times 180^\circ

A = \frac{\pi R^2 E}{180^\circ}

Variables

Symbol	Description	Unit
$A$	Area of the spherical polygon or triangle	-
$R$	Radius of the sphere	-
$E$	Spherical excess in degrees	-
$s$	Sum of interior angles in degrees	-
$n$	Number of sides	-

Terrestrial Sphere Applications

A major engineering application of spherical geometry is calculating distances along the Earth's surface. The Earth is modeled as a sphere (approximate radius $R \approx 6371\text{ km}$ or $3959\text{ miles}$ ).

Latitude: Defines small circles parallel to the equator.
Longitude: Defines great semicircles (meridians) intersecting at the poles.
Great Circle Distance: The shortest distance between two points on the surface of a sphere, calculated along the great circle connecting them using the central angle.

Great Circle Distance (Arc Length)

The shortest distance between two points on a sphere, where $\theta$ is the central angle in radians.

d = R \theta

Variables

Symbol	Description	Unit
$d$	Distance (arc length)	-
$R$	Radius of the sphere (e.g., Earth)	-
$\theta$	Central angle between the two points in radians	-

Spherical Pyramid

A solid bounded by a spherical polygon and the planes of the great circles forming the polygon's sides, all intersecting at the center of the sphere.

Volume of a Spherical Pyramid

The volume of a solid bounded by a spherical polygon and the planes of the great circles forming the polygon's sides, all intersecting at the center of the sphere.

V = \frac{1}{3} A_{base} \cdot R

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_{base}$	Area of the spherical polygon base	-
$R$	Radius of the sphere	-

Spherical Segment

A solid cut from a sphere by two parallel planes. If it is cut by one plane (a 'cap'), one radius is zero.

Volume of a Spherical Segment

The volume of a solid cut from a sphere by two parallel planes.

V = \frac{\pi h}{6} (3a^2 + 3b^2 + h^2)

Variables

Symbol	Description	Unit
$V$	Volume	-
$a$	Radius of lower base	-
$b$	Radius of upper base	-
$h$	Height of segment	-

Spherical Cap

A specific case of a spherical segment where one base is the curved surface of the sphere itself. It is cut by a single plane.

Volume and Surface Area of a Spherical Cap

The volume and curved surface area of a spherical cap.

V = \frac{\pi h^2}{3} (3R - h) = \frac{\pi h}{6} (3r^2 + h^2)

A_c = 2\pi R h = \pi(r^2 + h^2)

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_c$	Curved Surface Area	-
$R$	Radius of the original sphere	-
$r$	Radius of the circular base of the cap	-
$h$	Height of the cap	-

Interactive Simulation

Use the simulation below to explore the dimensions and properties of a spherical cap.

Spherical Zone

The portion of the surface of a sphere included between two parallel planes.

Area of a Spherical Zone

The area of a spherical zone is equal to the circumference of a great circle of the sphere multiplied by the height of the zone.

A_Z = 2\pi R h

Variables

Symbol	Description	Unit
$A_Z$	Area of the Zone	-
$R$	Radius of the sphere	-
$h$	Height of the zone	-

Spherical Sector

A solid bounded by a spherical zone and one or two conical surfaces.

Volume and Total Surface Area of a Spherical Sector

The total surface area of a spherical sector must include the area of the spherical zone ( $A_Z$ ) plus the lateral areas of any conical bounding surfaces.

V = \frac{1}{3} A_Z \cdot R = \frac{2}{3} \pi R^2 h

A_T = A_Z + \sum A_L

Variables

Symbol	Description	Unit
$V$	Volume of Sector	-
$A_T$	Total Surface Area	-
$A_Z$	Area of the spherical zone ( $2\pi R h$ )	-
$A_L$	Lateral area of the bounding cone(s)	-
$R$	Radius of the sphere	-

Spherical Wedge

A volume bounded by two intersecting planes and the spherical surface.

Spherical Lune

The surface area bounded by two great semicircles.

Spherical Wedge and Lune

Formulas for calculating the volume of a spherical wedge and the surface area of a spherical lune.

A_{lune} = 2 R^2 \theta

V_{wedge} = \frac{2}{3} R^3 \theta

Variables

Symbol	Description	Unit
$V_{wedge}$	Volume of Spherical Wedge	-
$A_{lune}$	Area of Spherical Lune	-
$R$	Radius of the sphere	-
$\theta$	Angle between the planes in radians	-

Key Takeaways

Spherical Geometry Elements: Spherical polygons have spherical excess ( $E$ ). A spherical triangle's angle sum must be between $180^\circ$ and $540^\circ$ .
Terrestrial Calculations: Distances between cities or coordinates are calculated as arc lengths of great circles ( $d = R\theta$ ).
Spherical Sectors: The total surface area of a spherical sector requires adding the spherical zone's area to the conical lateral area(s).
Archimedes' Relationship: Cone:Sphere:Cylinder volumes follow a 1:2:3 ratio when inscribed/circumscribed.

Previous TopicCylinders and Cones - Examples & Applications

Quiz Me

Next TopicSpheres and Spherical Geometry - Examples & Applications

Prev Next

Quiz Me