Have you ever wanted (or had) to know how to calculate the volume of your PC? Or even wondered what the surface area of your doughnut is?

**Definitions**

Throughout this entry:

**S** is the surface area of an object**V** is its volume**Π** represents Pi

**The Cuboid**

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

**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 = 2×12 + 2×32 + 2×24

S = 24 + 64 + 48

S = 136cm^{2}

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

*V = lbh
*

V = 3 × 4 × 8

V = 96

**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: 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 = 6l*^{2}

As its name implies, you can calculate the volume of a cube by cubing the length of one of its edges.

*V = l*^{3}

**The 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. 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.

**a** is the cross-sectional area of the prism**l** is the length of the prism^{1}

*V = al*

**The Cylinder**

A cylinder is a prism with 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.

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

Since the cylinder is formed from two identical circles and a rectangle, as stated earlier, all we need to do is calculate the areas of each of them and add them together:

*S = (Πr*^{2}) + (Πr^{2}) + (2Πr × l)

S = 2Πr^{2} + 2Πrl

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

*V = (l × Πr*^{2})

V = lΠr^{2}

**The 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.

**r** is the radius of the sphere

Surface Area:* S = 4Πr*^{2}

Volume:* V = 4/3Πr*^{3}

**The Torus**

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

**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

It sounds somehow funny, but the formula for the surface area of the torus is just like calculating the surface of the cylinder before it is twisted round into a circle (without the top and the bottom, of course). Basically, it's the circumference of the torus through the centre of the cylinder multiplied by the circumference of the cylinder, therefore:

*S = 2Π(a-b) x 2Πb*

or, simplified:

*S = 4b(a-b) Π*^{2}

The same principle applies for the volume of the torus; it's the circumference of the torus through the centre of the cylinder (that's the length of the cylinder before it is twisted into a torus) multiplied by the cross-sectional area of the cylinder.

*V = 2Π(a-b) × Πb*^{2}

or, simplified:

*V = 2(a-b)(Πb)*^{2}

**The 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.

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

**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 + Πr*^{2}

**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 = w*^{2} + 0.5lw + 0.5lw + 0.5lw + 0.5lw

S = w^{2} + 2lw

The equation for calculating the volume of a cone is the very same for all cones, no matter whether they have a circular or polygonal base.

**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*