The Beauty of Geometric Solids: An Introduction

Another famous shape named after Fuller is the carbon molecule (C₆₀) 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 r is the radius of the sphere. The diameter of a sphere is twice the length of its radius.

The volume of a geometric solid is the amount of space contained within the figure, while the surface area of a geometric solid is the extent of the figure's outside, or skin.

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 = 4/3πr³
where r is the sphere's radius, and π is approximately 3.14159.

The surface area of a sphere is calculated by the formula:
A = 4Πr²

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 * 3,959² = 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) * (Πr²), which can be written as:
V = 2 * Π² * R * r²

For a torus having r = 3 inches and R = 7 inches
V = 2 * Π² * 7 * 3²
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 * Π² * R * r
If we use the same dimensions as we did for the volume, we get:
A = 4 * Π² * 7 * 3
A ≈ 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 * r² * 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 ≈ 31.8 cubic inches.

The surface area of a cylinder is the sum of the surface area of both its ends, which is:
2 * π * r²
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 ≈ 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 * r² * 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 ≈ 29.32 cubic inches.

The surface area of a cone is determined by adding the area of the base, which is:
π * r²
and the area of the sides of the cone, which is:
π * r * s
where s is the 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
The surface area of the side is: 
A = π * 2 * √(2² + 7²)
A = π * 2 * √(4 + 49)
A = 2π√(53)
A ≈ 45.74
A = 12.57 + 45.74 ≈ 58.31 square inches.

If 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 * 5
V = 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 * 4
A = 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 ≈ 7.54 cubic inches.

The 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.

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.

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.

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.

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
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!