Ever wanted to know how to calculate the volume of your PC? Or even wondered what the surface area of your donut is?

Variables

Throughout this article:

S is the surface area of an object.
V is its volume.
π represents Pi.

The Cuboid

What is a cuboid?

A cuboid is a shape all the sides of which are squares or rectangles, like a matchbox. All cuboids have six faces, each of which has four edges.

Calculating the Surface Area

l is the length of the cuboid.
b is its breadth.
h is its height.

Calculating the surface area of a cuboid is very simple. Since there are six faces you can calculate the area of each face and then add them together:

S = lb + lb + lh + lh + hb + hb

S = 2lb + 2lh + 2hb

So if you had a cuboid measuring 3cm by 4cm by 8cm, you would calculate the surface area thus:

S = (2 × 3 × 4) + (2 × 4 × 8) + (2 × 3 × 8)

S = 24 + 64 + 48

S = 136cm²

Calculating the Volume

This is far easier than calculating the surface area. It's simply the height multiplied by the length multiplied by the width.

V = lbh

Easy, huh?

Cubes

A cube is a very simple form of cuboid which has edges that are all the same length. All the faces are therefore identical squares. Dice are good examples of cubes.

l is the length of one edge of the cube.

Surface Area of a Cube

Because all the faces are identical, and there are six of them, all we need to do is to calculate the area of one of the faces, and multiply by 6:

S = 6l²

Surface Area of a Cube

This solid is called a cube because you can find its volume by cubing the length of one of its edges.

V = l³

The Prism

What is a prism?

Any shape that has the same shape and area of cross-section all the way through is a prism. Therefore, a cylinder is a type of prism, as is a cuboid. Because all prisms are different in cross-section, each has a different equation for calculating their surface area.

Calculating the Volume

a is the cross-sectional area of the prism.
l is the length of the prism. Remember, a coin is a prism, and the length of that prism is the thickness of the coin.

The equation for calculating the volume of a prism, however, is constant. It is the cross-sectional area of the prism multiplied by its length.

V = al

A good example of a prism, as mentioned above, is...

The Cylinder

What is a cylinder?

A cylinder is a type of prism, one which has a circular cross-section. It is a very simple object, and can be cut (for mathematical purposes) into three polygons: two circles and a rectangle wrapped around them.

Calculating the Surface Area

r is the radius of the cylinder.
l is its length.

Since the cylinder is formed, as stated earlier, all we need to do is calculate the areas of each of them and add them together:

S = (πr²) + (πr²) + (2πr × l)

S = 2πr² + 2πrl

Calculating the Volume

Because a cylinder is a prism, calculating the volume is very simple. It is the cross-sectional area (i.e. the circle either at the top or bottom) multiplied by the height.

V = (l × πr²)

V = lπr²

The Sphere

What is a sphere?

A sphere is an object shaped like a tennis ball. It looks circular when viewed from any direction. This is a very strange object, mathematically, because it is so complex while being extremely simple.

Calculating the Surface Area

r is the radius of the sphere.

S = 4πr²

Calculating the Volume

V = 4/3πr³

The Torus

This kept me puzzled for quite a while...

What is a torus?

A torus is a 3D shape like a ring donut. It is formed of a cylinder twisted round into a circle.

Calculating the Surface Area

a is the radius of the entire torus.
b is the radius of the cylinder (the basic shape before it is twisted into a circle).

Basically, the formula is for the circumference of the torus through the centre of the cylinder multiplied by the circumference of the cylinder.

Therefore:

S = 2π(a−b) × 2πb

Tidied up a bit, we get:

S = 4b(a−b)π²

Calculating the Volume

This time, the formula is for the circumference of the torus through the centre of the cylinder multiplied by the cross-sectional area of the cylinder.

V = 2π(a−b) × πb²

So:

V = 2(a−b)(πb)²

The Cone

What is a Cone?

A cone is any object that tapers to a point (or apex). So a pyramid is a type of cone, as is a similar object with a 5, 7, or even 9 sided base.

Calculating the Surface Area

As for a prism, there are many different cones, so there are many different formulae for calculating the surface area. However, I will give the formulae for the standard cone, with a circular base, and for a pyramid, with a square base.

The simple cone's surface area

r is the radius of the cone.
l is the distance from the edge of the base of the cone to the apex.

The equation is very simple:

S = πrl + πr²

The pyramid's surface area

w is the length of one edge of the base of the pyramid.
l is the distance between the centre of one edge of the base and the apex of the pyramid.

Because the pyramid can be broken down into a square and four identical triangles, all we need to do is to calculate the area of each of these components and then add them together.

S = w² + 0.5lw + 0.5lw + 0.5lw + 0.5lw

S = w² + 2lw

Calculating the Volume

h is the distance from the centre of the base to the apex of the cone.
b is the area of the base

This equation is the same for all cones:

V = hb ÷ 3

Thanks to...

Gnomon,
manolan,
Amator,
Lucinda (AKA MyRedDice),
Dr St Justin, Patron Saint of Paper-Cuts,
Cefpret

Comments, Corrections and Suggestions

If you know how to calculate the volume/surface area of an interesting shape, like a rugby ball for instance, and I haven't mentioned it, I would value your input.

If I've made any massive blunders, or if you have something to add, or even if you just think I could have described anything here more simply, please tell me in the forum.