# The Beauty of Geometric Solids: An Introduction

If you've ever visited Walt Disney World in Florida, you've undoubtedly seen the geodesic dome called Spaceship Earth at Epcot. It is named after one of the terms made famous by American architect, Buckminster Fuller; a term which expressed his view of the world and its resources.

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It was Fuller who popularized the geodesic dome as an architectural feature. The shape is based on *geodesic polyhedra*, which are a class of geometric solid. Geodesic polyhedra are convex polyhedra made up of triangles. The usually have icosahedral symmetry, being made up of 20 equilateral triangular faces arranged around the surface of a sphere.

Another famous shape named after Fuller is the carbon molecule (C_{60}) buckminsterfullerene, which is in the shape of a truncated Icosahedron that resembles a soccer ball. It is made of **20** hexagons (a **6-sided** polygon) and **12** pentagons (a **5-sided** polygon).

Three scientists, Harold Kroto, Robert Curl, and Richard Smalley, were awarded the 1996 Novel Prize in Chemistry for their discovery of the class of fullerenes, which include buckminsterfullerene.

Geometric solids can be broken down into two classes: **Polyhedra** and **Non-Polyhedra**. Polyhedra have flat *faces*, or sides, and examples include cubes and pyramids. Non-Polyhedra don't have flat faces, and examples include the sphere, cylinder, torus, and cone. Let's examine the Non-Polyhedra first.

### Sphere

Like its 2D counterpart, the circle, a sphere is defined as the set of points, in 3-dimensional space, that are the same distance * r* from a given point (the center), where

*is the radius of the sphere. The*

**r***of a sphere is twice the length of its radius.*

**diameter**The * volume* of a geometric solid is the amount of space contained within the figure, while the

**of a geometric solid is the extent of the figure's outside, or skin.**

*surface area*Of all the geometric solids, a sphere has the smallest surface area for a given volume. Nature takes advantage of this property in the formation of water droplets and bubbles.

The volume of a sphere is determined by the formula:

**V = 4/3πr ^{3}**

where

**is the sphere's radius, and**

*r***π**is approximately

**3.14159**.

The surface area of a sphere is calculated by the formula:

**A = 4Πr ^{2}**

As an example, the radius of the Earth is **3,959 miles** (**6,378 km**), so we can calculate the surface area of the Earth as:

**A = 4 * Π * 3,959 ^{2} = 196,961,118 square miles**.

Since **71%** of the Earth's surface is ocean, that leaves us with **57,118,725 square miles** on which to live.

In reality, the Earth is not a sphere, but a *spheroid*, that is, it is slightly flattened at the poles. The Earth's polar radius is **3,950 miles** (**6,357 km**), while its equatorial radius is **3,963 miles** (**6,378 km**).

The Earth is an *oblate spheroid*, while the familiar American football is a *prolate spheroid*. One half of a sphere is called a *hemisphere*, and on Earth, from the north pole to the equator is the Northern Hemisphere, and from the equator to the South Pole is the Southern Hemisphere.

### Torus

To describe a torus, think of the shape of a donut or an inner tube. A torus is defined by two radii: **r**, which is the radius of a small circle that is revolved along a line made by a larger circle that has radius **R**.

To find the volume of a torus, we have to take into account both radii:

**V = (2ΠR) * (Πr ^{2})**, which can be written as:

**V = 2 * Π**

^{2}* R * r^{2}For a torus having **r** = **3 inches** and **R** = **7 inches**

**V = 2 * Π ^{2} * 7 * 3^{2}**

V ≈ 1,244 cubic inches

The surface area of a torus is determined by the formula:

**A = (2ΠR) * (2Πr)**, which can be written as:**A = 4 * Π ^{2} * R * r**If we use the same dimensions as we did for the volume, we get:

**A = 4 * Π**

A ≈ 829 square inches

^{2}* 7 * 3A ≈ 829 square inches

### Cylinder

Cylinders are familiar to us from canned goods, which come in cylinders. Cylinders come in two general types: **Right** and **Oblique**. If the two ends of a cylinder are aligned with one another, it is considered a **Right Cylinder**, otherwise, it is an **Oblique Cylinder**.

The volume of a cylinder is determined by the area of its base times its height:

**V = Π * r ^{2} * h**So, for a can of baked beans which has a radius of

**1.5 inches**and a height of

**4.5 inches**, its volume is:

**V = 3.14159 * 2.25 sq in * 4.5 in**

V.

V

**≈**31.8 cubic inchesThe surface area of a cylinder is the sum of the surface area of both its ends, which is:

**2 * π * r ^{2}**

plus the surface area of the sides, which is:

**2 * π * r * h**

Therefore, the total surface area of a cylinder is:

**A = 2 * Π * r * (r + h)**

For our can of baked beans:

**A = 2 * Π * 1.5 * 6**

A.

A

**≈**56.5 square inches### Cone

A cone is a geometric solid that has a circle at one end, called the *base*, and a point at the other end, called the *apex*. As with cylinders, when the apex is aligned with the center of the base, the cone is a called a **Right Cone**, otherwise it is called an **Oblique Cone**.

The volume of a cone is determined by the radius of its base and the height of its apex:

**V = 1/3 Π * r ^{2} * h**

An average waffle-type ice cream cone has a radius of

**2 inches**and a height of

**7 inches**. To find out the volume ice cream it can hold:

**V = 1/3 * 3.14159 * 4 sq in * 7 in**

V.

V

**≈**29.32 cubic inchesThe surface area of a cone is determined by adding the area of the base, which is:**π**** * r ^{2}**

and the area of the sides of the cone, which is:

**π**

*** r * s**

where

**is the**

*s**slant length*, which is the distance from the base to the apex measured along the object's side.

Therefore the surface area of a cone is:

**A =**

**π*** r * (r + s)For a cone having

**r = 2**and

**h = 7**, the surface area of the base would be:

**A = 3.14159 * 4**

A ≈ 12.57

A ≈ 12.57

The surface area of the side is:

**A = π * 2 * √(2**

^{2 }+ 7^{2})**A =**

A = 2π√(53)

A ≈ 45.74

A = 12.57 + 45.74.

**π*** 2 * √(4 + 49)A = 2π√(53)

A ≈ 45.74

A = 12.57 + 45.74

**≈**58.31 square inchesIf we compare the volume of a cylinder and a cone that have the same size base and height, the volume of the cone is exactly **1/3** that of the cylinder. That means if ice cream cones came in cylinders and not cones, you'd get three times as much ice cream. Yay!

### Polyhedrons

Now that we've examined the Non-Polyhedron geometric solids, it's time to take a look at the Polyhedron solids. A *polyhedron* is a geometric solid that has flat faces, or *polygons*, which are 2D figures having at least **3** straight sides and angles. In Greek, poly means "many" and hedron means "face".

The main types of polyhedra are:

- Cuboids and cubes
- Platonic solids
- Prisms
- Pyramids

### Cuboids and Cubes

Cuboids are box-shaped objects that have **6 flat faces**, and all their angles are right, or **90 °** angles. Cuboids have a *length*, a *width*, and a *height*. When all three (length, width and height) are the same, a cuboid is called a cube, and each of its faces is a square. A cube has **6 faces**, **8 verticies** and **12 edges**.

We determine the volume of a cuboid by:

**V = length * width * height**

So, for a box having a length of **10 inches**, a width of **4 inches**, and a height of **5 inches:V = 10 * 4 * 5V = **

**200 cubic inches**.

That's good to know if you want to ship a package.

The surface area of a cuboid is determined by:

**A = 2 * width * length + 2 * length * height + 2 * height * width**

For the box having a length of **10 inches**, a width of **4 inches**, and a height of **5 inches**:

**A = 2 * 4 * 10 + 2 * 10 * 5 + 2 * 5 * 4A = 220 square inches**.

This is also good to know if you want to wrap a box.

### The Platonic Solids

Named for the ancient Greek philosopher Plato, these are 3D shapes where each face is a *regular polygon*, that is, a polygon whose sides are all the same length. Also, a Platonic Solid must have the same number of polygons meeting at each *vertex,* or corner. That means that the cube we just met above is a Platonic Solid, because each of its faces is a same-sized square, and **3 squares** meet at each of its vertices..

### Tetrahedron

Another Platonic Solid is the Tetrahedron, which is also known as a triangular pyramid. It is comprised of **4 triangular faces**, **6 straight edges** and **4 vertices**. It is the only Platonic Solid that has no parallel faces, and is the simplest of all the Platonic Solids.

When a Tetrahedron has all faces the same size and shape, it is a **Regular Tetrahedron**, otherwise it is an **Irregular Tetrahedron**.

The volume of a Tetrahedron is determined by:

**V = √2/12 * (Edge Length) ^{3}**

For a tetrahedron having an edge length of

**4 inches**

**V = 1.414/12 * 64**

V.

V

**≈**7.54 cubic inchesThe surface area of a tetrahedron can be found by:

**A = √3 * (Edge Length) ^{2}**

so, for our tetrahedron having an edge length of

**4**, its surface area would be:

**A = 1.732 * 16**

A = ≈ 27.71 square inches.

A = ≈ 27.71 square inches

### Octahedron

An Octahedron is like two square pyramids connected at their bases. It has **4** triangles that meet at each vertex, **8** **faces**, **6** **vertices** and **12 edges**.

We can calculate the volume of an octahedron by:

**V = (√2)/3 * (Edge Length) ^{3}**

For an octahedron having an edge length of

**4 inches**, its volume would be:

**V = 1.414 / 3 * 64**

V ≈ 30.17 cubic inches.

V ≈ 30.17 cubic inches

The surface area of an octahedron is:

**A = 2 * √3 * (Edge Length) ^{2} A = 2 * 1.732 * 16 A ≈ 55.42 square inches**.

### Dodecahedron

This Platonic Solid is formed when **3 pentagons** (**5-sided** polygons) meet at each vertex, it has **12 faces**, **20 vertices** and **30 edges**. A Dodecahedron gets its name from the Greek *dodeca*, which means 12.

The volume of a Dodecahedron is:**V = (15 + 7 * √5)/4 * (Edge Length) ^{3}**

For a Dodecahedron having an edge length of

**4 inches**, it's volume would be:

**V = (15 + 7 * 2.236) / 4 * 64**

V ≈ 490.43 cubic inches.

V ≈ 490.43 cubic inches

The formula for finding the surface area of a Dodecahedron is:

**A = 3 * √(25 + 10 * √5) * (Edge Length) ^{2} A = 3(25 + 22.36) * 16 A ≈ 330.33 square inches**.

### Icosahedron

The most complex of the Platonic Solids, at each of its vertices, **5 trangles** meet, the Icosahedron has **20 faces** each of which is an equilateral triangle (a triangle having **3 equal** sides and **3 equal** angles of **60°**), **12 vertices** and **30 edges**.

The Icosahedron might be familiar to you from playing games that use 20-sided dice, and mother nature apparently also has a fondness for this shape, because the outer shell of the human papilloma virus is an Icosahedron.

The volume of an Icosahedron is determined by the formula:

**V = 5 * (3 + √5)/12 * (Edge Length) ^{3}**

so, for an Icosahedron having an edge length of

**4 inches**, its volume would be:

**V = 5(5.236) / 12 * 64**

V ≈ 139.63 cubic inches.

V ≈ 139.63 cubic inches

The formula for calculating the surface area of an Icosahedron is:

**A = 5 * √3 * (Edge Length) ^{2} A ≈ 138.56 square inches**.

### Prisms

A prism is a geometric solid having identical ends, flat faces, and the same cross section along its length. The two ends of a prisim are called its *bases*, and the faces of a prism are all *parallelograms* (a 2D figure whose opposite sides are parallel and equal, and whose opposite angles are equal).

By this definition, the cuboid and cubes we met above are prisms, but you can also have triangular, pentagonal, and hexagonal prisms, whose cross sections are a triangle, pentagon and hexagon, respectively.

The cross sections of **Regular Prisms** have equal edge lengths and equal angles, while the cross sections of **Irregular Prisms** have unequal edge lengths and unequal angles.

If the bases of a prism are aligned with each other, the prism is said to be a **Right Prism**, if the bases are not aligned with one another, it is said to be an **Oblique Prism**.

We can determine the volume of a prism by:

**Volume = Base Area * Length**

For a triangular prism having a base area of **25 square inches** and a length of **10 inches**, its volume would be:

**V = 25 sq in * 10 inches V = 250 cubic inches.**

We can find the surface area of a triangular prism by: **2 * Base Area + Base Perimeter * Length**

If we use the example from above, our triangular prism has a base area of **25 square inches**, a length of **10 inches**, and a base perimeter of **24 inches**:

**A = 2 * 25 square inches + 24 inches * 10 inches A = 290 square inches**

### Pyramids

A pyramid is defined by having a base that is a polygon, an apex, and faces that are triangles. The famous pyramids on Egypt's Giza Plateau are actually **Square Pyramids** because their bases are a square. You can also have a pyramid with a triangular base called a Triangular Pyramid, and a pyramid with a pentagon as its base called a Pentagonal Pyramid.

If a pyramid's apex is directly over the center of its base, it is said to be a **Right Pyramid**. If the apex isn't over the center of the base, it is said to be an **Oblique Pyramid**.

The volume of a pyramid is determined by:

**V = 1/3 * Base Area * height**Let's determine the volume of the Pyramid of Khufu, the largest of the three Giza Plateau pyramids. The length of each side of its base is

**756 feet**or

**230.34 meters**. Therefore, its base area is

**571,536**

**square feet**or

**53,056.5**

**square meters**. The height of the Great Pyramid is

**455 feet**or

**138.7 meters**, therefore the volume of the Great Pyramid is:

**V = 1/3 * 571,536 sq. ft. * 455 feet**

V = 86,682,960 cubic feet

V = 86,682,960 cubic feet

That's a whole lot of room for Pharaoh Khufu, who is buried in the pyramid.

The surface area of a pyramid has two parts: the **Base Area** and the **Lateral Area**. For an Irregular Pyramid, you must add together the area of each of its triangular faces to find its surface area, but for a Regular Pyramid, we can find the Lateral Area by:

**A = (Perimeter * Slant Length) / 2**

For the Great Pyramid whose base length is **756 feet**, its perimeter is **3,024 feet** and its Slant Length is **612 feet** or **186.42 meters**. Therefore, the Lateral Surface Area of the Great Pyramid is:

**A = (3,024 * 612) / 2**

which is **925,344 square feet**.

### Hundreds of geometric solids

There are well over **100** other geometric solids whose beauty is undeniable, and you can see them in action, rotating in 3-space, at the website Math is Fun. Enjoy!