Polyhedra and Prisms - Theory & Concepts

Polyhedra Formulas

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 related by Euler's Formula.

V - E + F = 2

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: $\approx 138.19^\circ$

Platonic Solids

Select Shape

Euler's Formula Properties

Vertices (V)4

Edges (E)6

Faces (F)4

Euler's Formula: V - E + F = 2

4 - 6 + 4 = 2

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}

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}

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}}

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}

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

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

Prism Properties

Length (l)4.0

Width (w)3.0

Height (h)5.0

Base Area12.0

Perimeter14.0

Volume (V)60.00

Surface Area (A)94.00

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

Volume of a Rectangular Parallelepiped

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

V = l \cdot w \cdot h

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)

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}

Volume of a Cube

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

V = s^3

Surface Area of a Cube

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

A = 6s^2

Space Diagonal of a Cube

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

d = s\sqrt{3}

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}

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

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

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)

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.

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