Platonic Solids

The Platonic solids are a set of five regular polyhedra - the tetrahedron, cube, octahedron, dodecahedron, and icosahedron - with equivalent faces composed of congruent convex regular polygons. Each of the five Platonic solids depends upon a two-dimensional shape: The tetrahedron, octahedron, and icosahedron's faces are based on triangles; the cube's faces are based on squares, and the dodecahedron has pentagonal faces.

Concepts to review before learning Platonic solids

Before diving into Platonic solids, there are a few concepts to make sure you're solid on, like polyhedra, pyramids, and cross sections. It's also a good idea to review how to find the area of a polygon, as you'll need this to find the surface area and volume of a Platonic solid.

An overview of Platonic solids

Each of the Platonic solids has faces, edges, and vertices. When finding the surface area or volume of a Platonic solid, you will need to know the measurement of the edge. Luckily, all of the edges of a Platonic solid are the same. Let's take a look at the different Platonic solids and how to find the surface area and volume of each.

The tetrahedron

The tetrahedron has 4 faces, each of which is an equilateral triangle. It has 6 edges and 4 vertices. At each vertex, 3 edges meet.

We can find the surface area and volume of a tetrahedron if we know the length of its edge (e). For this tetrahedron, $e=6\mathrm{cm}$ . To find its surface area (A), we need to plug in the length to the formula:

$A=\sqrt{3}\cdot e^2$

$A=\sqrt{3}\cdot 6^2=62.354{\mathrm{cm}}^2$

We use a slightly different formula to find the tetrahedron's volume (V):

$V=\frac{e^3}{6\sqrt{2}}$

$V=\frac{6^3}{6\sqrt{2}}=25.456{\mathrm{cm}}^3$

The cube

The cube has 6 faces. Each face is a square. It has 12 edges and 8 vertices. At each vertex, 3 edges meet.

We can find the surface area and volume of a cube if we know the length of its edge (e). For this cube, e = 12 in. To find its surface area (A), we will plug in the length to the formula:

$A=6e^2$

$A=6{\left(12\right)}^2=684{\mathrm{in}}^2$

To find the volume (V) of a cube, we will use this formula:

$V=e^3$

$V={12}^3=1728{\mathrm{in}}^3$

The octahedron

The octahedron has 8 faces. Each is an equilateral triangle. It has 12 edges and 6 vertices. At each vertex, 4 edges meet.

We can find the surface area and volume of an octahedron if we know the length of its edge (e). For this example, e = 7.5 mm. To find its surface area (A), we will plug in the length to the formula:

$A=2\sqrt{3}\cdot e^2$

$A=2\sqrt{3}{\left(7.5\right)}^2=194.86{\mathrm{mm}}^2$

To find the volume (V) of an octahedron, we'll use the formula:

$V=\frac{\sqrt{2}}{3e^3}$

$V=\frac{\sqrt{2}}{3{\left(7.5\right)}^3}=198.87{\mathrm{mm}}^3$

The dodecahedron

The dodecahedron has 12 faces. Each is a regular pentagon. It has 30 edges and 20 vertices. At each vertex, 3 edges meet.

To find the surface area and volume of a dodecahedron, we need to know the length of its edge (e). For this example, $e=32$ ft. To find its surface area (A), we will plug in the length to the formula:

$A=3\sqrt{25+10\sqrt{5}}\cdot e^2$

$A=3\sqrt{25+10\sqrt{5}}\cdot {32}^2=21141.23{\mathrm{ft}}^2$

To find the dodecahedron's volume, use:

$V=15+\frac{7\sqrt{5}}{4e^3}$

$V=15+\frac{7\sqrt{5}}{4\cdot 3^3}=251105{\mathrm{ft}}^2$

The icosahedron

The icosahedron has 20 faces. Each is an equilateral triangle. It has 30 edges and 12 vertices. At each vertex, 5 edges meet.

To find the surface area and volume of an icosahedron, we need to know the length of its edge (e). For this example, $e=2\mathrm{m}$ . To find its surface area (A), we will plug in the length to the formula:

$A=5\sqrt{3}{e}^2$

$A=5\sqrt{3}{2}^2=34.64{\mathrm{m}}^2$

If you'd like to find an icosahedron's volume (V), use the formula:

$V=\frac{5\left(3+\sqrt{5}\right)}{12}{e}^3$

$V=\frac{5\left(3+\sqrt{5}\right)}{12}{2}^3=17.454{\mathrm{m}}^3$

Practice questions for Platonic solids

a. Find the volume (V) of a tetrahedron with an edge (e) of 14 cm.

$V=\frac{{14}^3}{6}\sqrt{2}=323.384{\mathrm{cm}}^3$

b. Find the surface area (A) of a cube with an edge (e) of 3.75 in.

$A=6{\left(3.75\right)}^2=84.38{\mathrm{in}}^2$

c. What is the volume (V) of an octahedron with an edge (e) of 192 mm?

$V=\frac{\sqrt{2}}{3}{192}^3=3.333655\times {10}^6{\mathrm{mm}}^3$

d. What is the surface area (A) of a dodecahedron with an edge (e) of 11 ft?

$A=3\sqrt{25+10\sqrt{5}}\times {11}^2=2498.13{\mathrm{ft}}^2$

e. What is the volume (V) of an icosahedron with an edge (e) of 4.33 meters?

$V=\frac{5(3+\sqrt{5})}{12}\times {4.33}^3=177.12{\mathrm{m}}^3$

Topics related to the Platonic Solids

Polyhedra

Pyramid

Cross Sections

Flashcards covering the Platonic Solids

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Platonic Solids

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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