Polyhedra and Prisms - Theory & Concepts

Polyhedra Formulas

Learning Objectives

Classify and understand properties of polyhedra and regular polyhedra (Platonic Solids).
Apply Euler's Formula to analyze the vertices, edges, and faces of convex polyhedra.
Calculate volume and surface area for standard prisms, including general, right, and truncated prisms.
Compute volume and surface area for rectangular parallelepipeds and cubes.
Determine volume and surface properties of general and regular pyramids, as well as their frustums.

Polyhedra are 3D solid figures with flat polygonal faces, straight edges, and sharp corners (vertices). Important concepts include Euler's Formula and Platonic Solids.

Euler's Formula for Polyhedra

For any convex polyhedron, the number of vertices $V$ , edges $E$ , and faces $F$ are fundamentally related by Euler's Formula. This relationship is a cornerstone of discrete geometry.

Euler's Polyhedral Formula

Relates the number of vertices, edges, and faces of a convex polyhedron.

V - E + F = 2

Variables

Symbol	Description	Unit
$V$	Number of vertices (corners)	-
$E$	Number of edges	-
$F$	Number of faces	-

Platonic Solids (Regular Polyhedra)

A regular polyhedron is one whose faces are identical regular polygons, and the same number of faces meet at each vertex. There are exactly five Platonic solids, each with a specific interior dihedral angle:

Tetrahedron: 4 equilateral triangle faces. Dihedral angle: $\approx 70.53^\circ$
Hexahedron (Cube): 6 square faces. Dihedral angle: $90^\circ$
Octahedron: 8 equilateral triangle faces. Dihedral angle: $\approx 109.47^\circ$
Dodecahedron: 12 regular pentagon faces. Dihedral angle: $\approx 116.57^\circ$
Icosahedron: 20 equilateral triangle faces. Dihedral angle: $\approx 138.19^\circ$

Interactive Simulation

Use the simulation below to explore the properties and shapes of the five Platonic solids.

Volume and Surface Area of a Regular Tetrahedron

A regular tetrahedron is a polyhedron composed of four equilateral triangles. Its volume and surface area can be derived directly from its edge length.

V = \frac{a^3 \sqrt{2}}{12}

A = a^2 \sqrt{3}

Variables

Symbol	Description	Unit
$V$	Volume	-
$A$	Total Surface Area	-
$a$	Edge length	-

Volume and Surface Area of a Regular Octahedron

A regular octahedron is composed of eight equilateral triangles. Its volume and surface area can be determined from its edge length.

V = \frac{a^3 \sqrt{2}}{3}

A = 2 a^2 \sqrt{3}

Variables

Symbol	Description	Unit
$V$	Volume	-
$A$	Total Surface Area	-
$a$	Edge length	-

Volume and Surface Area of a Regular Dodecahedron

A regular dodecahedron is composed of twelve regular pentagons.

V = \frac{a^3}{4} (15 + 7\sqrt{5})

A = 3 a^2 \sqrt{25 + 10\sqrt{5}}

Variables

Symbol	Description	Unit
$V$	Volume	-
$A$	Total Surface Area	-
$a$	Edge length	-

Volume and Surface Area of a Regular Icosahedron

A regular icosahedron is composed of twenty equilateral triangles.

V = \frac{5a^3}{12} (3 + \sqrt{5})

A = 5 a^2 \sqrt{3}

Variables

Symbol	Description	Unit
$V$	Volume	-
$A$	Total Surface Area	-
$a$	Edge length	-

Regular Prism

A right prism whose bases are regular polygons. Because it is a right prism, its lateral faces are all identical rectangles. This uniformity simplifies calculations for perimeter, base area, and lateral area.

Volume of a General Prism

The volume of any prism (or cylinder) is the product of its base area and its perpendicular height.

V = A_b \cdot h

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_b$	Area of the base	-
$h$	Perpendicular height	-

Lateral Surface Area of a General Prism

The general formula for the lateral area of any prism relies on the perimeter of its right section and the length of its lateral edge.

A_L = P_R \cdot L_e

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$P_R$	Perimeter of the right section	-
$L_e$	Length of the lateral edge	-

Interactive Simulation

Use the simulation below to adjust prism parameters and observe changes in volume and area.

Surface Area of a Right Prism

For a right prism, the right section is identical to the base, and the lateral edge equals the perpendicular height. Thus, the lateral area is the perimeter of the base multiplied by the height. The total surface area adds the areas of the two bases.

A_L = P_b \cdot h

A = A_L + 2A_b

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$A$	Total Surface Area	-
$P_b$	Perimeter of the base	-
$h$	Perpendicular height	-
$A_b$	Area of the base	-

Volume of a Rectangular Parallelepiped

A prism with rectangular bases, commonly known as a rectangular box.

V = l \cdot w \cdot h

Variables

Symbol	Description	Unit
$V$	Volume	-
$l$	Length	-
$w$	Width	-
$h$	Height	-

Surface Area of a Rectangular Parallelepiped

The total surface area of a rectangular box is the sum of the areas of its six rectangular faces.

A = 2(lw + lh + wh)

Variables

Symbol	Description	Unit
$A$	Total Surface Area	-
$l$	Length	-
$w$	Width	-
$h$	Height	-

Space Diagonal of a Rectangular Parallelepiped

The space diagonal (or body diagonal) connects two opposite corners through the interior of the box.

d = \sqrt{l^2 + w^2 + h^2}

Variables

Symbol	Description	Unit
$d$	Space Diagonal	-
$l$	Length	-
$w$	Width	-
$h$	Height	-

Volume of a Cube

A cube is a special prism where all edges are equal length.

V = s^3

Variables

Symbol	Description	Unit
$V$	Volume	-
$s$	Side length	-

Surface Area of a Cube

The total surface area of a cube is six times the square of its side length.

A = 6s^2

Variables

Symbol	Description	Unit
$A$	Total surface area	-
$s$	Side length	-

Space Diagonal of a Cube

The space diagonal of a cube, connecting two opposite corners.

d = s\sqrt{3}

Variables

Symbol	Description	Unit
$d$	Space Diagonal	-
$s$	Side length	-

Volume of a Truncated Prism

A truncated prism is a portion of a prism cut off by a plane not parallel to the base. Its volume is the product of the right section area and the average length of its lateral edges.

V = A_R \cdot L_{avg}

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_R$	Area of the right section	-
$L_{avg}$	Average length of lateral edges	-

Volume of a General Pyramid

The volume of a pyramid is one-third the product of its base area and perpendicular height.

V = \frac{1}{3} A_b \cdot h

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_b$	Area of the base	-
$h$	Perpendicular height	-

Lateral Surface Area of a Regular Pyramid

For a regular pyramid (base is a regular polygon and the altitude passes through its center), the lateral area is half the product of its base perimeter and slant height.

A_L = \frac{1}{2} P_b \cdot L

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$P_b$	Perimeter of the base	-
$L$	Slant height	-

Volume of a Frustum of a Regular Pyramid

A frustum is the lower portion of a pyramid left after its top is cut off by a plane parallel to its base. For a regular pyramid, the formula relies on the areas of the upper and lower regular polygon bases and the perpendicular height between them.

V = \frac{h}{3} \left( A_1 + A_2 + \sqrt{A_1 A_2} \right)

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_1$	Area of the lower base	-
$A_2$	Area of the upper base	-
$h$	Perpendicular height between bases	-

Key Takeaways

Prisms: Volume is base area multiplied by height ( $V = A_b h$ ). The lateral area for general prisms relies on the right section ( $A_L = P_R \cdot L_e$ ).
Pyramids: Volume is exactly one-third of the enclosing prism ( $V = \frac{1}{3} A_b h$ ).
Diagonals: The space diagonal passes through the solid's interior, extending the Pythagorean theorem to 3D.

Previous TopicIntroduction to Solid Mensuration - Examples & Applications

Quiz Me

Next TopicPolyhedra and Prisms - Examples & Applications

Prev Next

Quiz Me