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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?
Throughout this entry:
- S is the surface area of an object
- V is its volume
- Π represents Pi
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 = 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
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
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
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
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
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
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
V = 2(a-b)(Πb)2
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