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Calculating the Volume and Surface Area of Various Solid Objects

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
• 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 = 136cm2

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 = 6l2

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

V = l3

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 prism1
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 = (Πr2) + (Πr2) + (2Πr × l)
S = 2Πr2 + 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 × Πr2)
V = lΠr2

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 = 4Πr2
Volume: V = 4/3Πr3

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) Π2

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) × Πb2

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 + 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

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

1 Remember, a coin is a prism, and the length of that prism is the thickness of the coin.

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Entry Data
Entry ID: A533189 (Edited)

Written and Researched by:
Peter aka Krans

Edited by:
Jeremy (trying to find his way back to dinner)

Date: 09   May   2001