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Mathematics LibreTexts

4.3: 3-D Geometry


Thinking out Loud

When you slice an orange, what type of shape can occur?

Polyhedra

Definition: Polyhedra

Polyhedra (pl.) are simple closed surfaces that are composed of polygonal regions.

A polyhedron (sg.) has a number of:

  • Vertices - corners where various edges and polygonal corners meet
  • Edges - lines where two polygonal edges meet
  • Faces - the proper name for polygonal regions which compose a polyhedron

Polyhedra may be:

  • Convex - shapes which follow the convex property of 2-dimensional geometry in 3-dimensional space.
  • Concave - shapes which follow the concave property of 2-dimensional geometry in 3-dimensional space.

Nets are used when constructing polyhedra out of a single, contiguous, piece of material. The various polygons are laid out together, edges touching, to be cut out and folded together.

Polyhedra are said to be prisms if they have bases which are congruent and parallel polygons

Pyramids

Definition: Pyramids

Pyramids are created by joining a polygonal base to a point above it, called an apex. Each edge of the base, when joined by its vertices to the apex, creates a series of congruent triangles around the shape.

The more edges the base has, the more a pyramid approximates a cone.

Thinking Out Loud:

The sphere is the most symmetrical solid in space. Building a sphere isn’t easy, so what other solids might we construct to approximate its symmetry? In order to construct a solid with lots of symmetry, we suppose our solid has flat sides and straight edges. What properties would such solids have in order to be symmetrical as possible?

PRISMS

Definition

 

Regular Polyhedra

Definition: Regular polyhedra (Platonic solids)

A polyhedron is said to be regular if:

  • All of its faces are congruent
  • All of its vertices join the same number of edges
  • All of its edges join only two faces

Regular polyhedra are also called the Platonic solids.

Thinking Out Loud:

How many regular polyhedra are possible? How can you prove it?

Tetrahedron

Definition: tetrahedron

A tetrahedron is a Platonic solid with:

  • 4 faces
  • 4 vertices
  • 6 edges

The tetrahedron is bounded by four equilateral triangles and has the smallest volume for its surface area of the Platonic solids. 

In Ancient Greece, a tetrahedron represents the property of dryness and corresponds to the element of Fire. 

Cube (hexahedron)

Definition: Cube

A cube, or hexahedron, is a Platonic solid with:

  • 6 faces
  • 8 vertices
  • 12 edges

The cube is bounded by six squares.

In Ancient Greece, the cube, standing firmly on its base, corresponds to the element of Earth.

Octahedron

Definition: Octahedron

An octahedron is a Platonic solid with:

  • 8 faces
  • 6 vertices
  • 12 edges

The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices.

In Ancient Greece, the octahedron corresponds to the element Air.

Dodecahedron

Definition: Dodecahedron

A dodecahedron is a Platonic solid with:

  • 12 faces
  • 20 vertices
  • 30 edges

The dodecahedron is bounded by twelve congruent regular pentagons.

In Ancient Greece, the dodecahedron corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.

Icosahedron

Definition: Icosahedron

An icosahedron is a Platonic solid with:

  • 20 faces
  • 12 vertices
  • 30 edges

The icosahedron is bounded by twenty equilateral triangles and has the largest volume for its surface area of the Platonic solids.

In Ancient Greece, the icosahedron represents the property of wetness and corresponds to the element of Water.

Thinking Out Loud:

Where in your life might you have seen the Platonic solids? How might the Platonic solids be useful?

Euler's Formula

Definition

There is a relationship between the number of faces (F), vertices (V), and edges (E) in any convex polyhedron, and knowing this relationship enables us to construct a formula that connects the number of faces, vertices, and edges.

Euler's formula for convex polyhedra is: \(V + F = E + 2\)

That is: for any convex polyhedron, the number of vertices added to the number of faces is equal to two more than the number of edges.

Example \(\PageIndex{1}\):

Consider the dodecahedron:

\(V = 20, \, F = 12, \, E = 30\).

Let's see if Euler's formula holds:

\(V + F = E + 2\)

\(V + F=(20) + (12)=32 \) and \(E + 2= (30) + 2=32\)

Excellent! It works.

Truncated Regular Polyhedra

Definition

Truncated regular polyhedra, which are also sometimes called archimedian solids, must:

  • Be composed of regular polygons
  • Have identical vertices
  • Not be a Platonic solid, prism, or anti-prism.

Example \(\PageIndex{2}\):

Non-Convex Uniform Polyhedra (Kepler-Poinsot Solids)

It is possible to construct regular polyhedra that are not convex - that is, shapes that have identical faces but also have incuts or void spaces. These solids are sometimes called Kepler-Poinsot polyhedra. These shapes can be made by building a regular dodecahedron or icosahedron and adding pyramidal or pentagramal volumes to each face. These polyhedra do not always satisfy the Euler relation as it relates to Platonic solids.

There are four Kepler-Poinsot polyhedra: three based on the dodecahedron and one built upon the icosahedron.

Example \(\PageIndex{3}\): Kepler-poinsot solids

Cylinders, Spheres, and Cones

Cylinders

By BR84 (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Cylinders are prisms with two circular faces. Their volume is given by

\[V_{cylinder} = \pi r^2h\]

and surface area by

\[S_{cylinder} = \left( 2 \pi r \right) \left( r + h \right) \]

where \(r\) is the radius of the circular faces, and h is the distance between the two circular faces of the cylinder, or its height. A cylinder does not need perfect circles as its bases, provided both are congruent. The most common alternative to a circular base is an elliptical one. A shape with this base would be called an elliptical cylinder.

Spheres

Spheres are perfectly round objects that are found in a 3-dimensional Euclidean space. Technically, they have no thickness and are hollow. The volume described by a sphere is given by

\[\displaystyle V_{sphere}= \dfrac{4}{3} \pi r^3\]

here \(r\) is the radius of the sphere. The surface area of a sphere is given by

\[S_{sphere} = 4 \pi r^2.\]

 

By Original: Andertxuman Vector version: CheCheDaWaff (This file was derived from  Esfera Arquímedes.jpg:) [Public domain], via Wikimedia Commons

Cones

Cones are pyramids with circular bases. Their volume is given by

\[V_{cone} = \dfrac{1}{3} A_Bh \]

where h is the height of the cone and AB is the area of the base. The surface area of a cone is described by the formula \(S = \pi r^2+ \pi r \surd \left( r^2 + h^2 \right)\)

 

Cylinder image  By BR84 (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons