Units of measurement and volume and surface area formulae

7/30/2019 Units of measurement and volume and surface area formulae

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Measurement

We seem to live in an eleven-dimensional space, most of whose dimensions are folded up into

something called the Calabi-Yau Manifold. Here is a section of a quintic Calabi-Yau threefoldprojected into three-dimensional space (though obviously then projected onto the virtuallytwo-dimensional surface of this monitor or piece of paper):

According to Doctor Who, various beings live in Calabi-Yau space, including the Guardians ofTime, Chronovores, the Great Old Ones (for example Nyarlathotep and Cthulhu) and I reckon

also the reapers and so on. I also sometimes wonder if an ex-friend of mine belongs there.

However, all of this can be safely ignored if you consider yourselves to be entities consisting ofa single world-line existing in space and time and having finite mass, as I expect you do. As far

as we're concerned for the purposes of this document, there are three dimensions of space,one of time and one of mass, and these are the things I'm going to talk about here.

Everything in a small region of space can be pretty accurately located at a particular moment

using three numbers to describe its position. For instance, my head is currently about a metrefrom the wall to my left, a metre and a half from the floor and three metres from the French

windows behind me. The fact that I only need three numbers to describe where my head is.


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Consider this humble toilet roll:

This has a location within this room which can be described using those three numbers, usingthe X, Y and Z axes:


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If you wanted to tell someone where that toilet roll was, you would only need three numbersto do it, and those numbers would represent measurements along those three axes.

The simplest measurement to describe is probably length. The metric system uses a unit

called the metre (often written as m), to measure length. This was originally defined asfollows. Here's Earth:


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If you imagine a line like this:

drawn from the North Pole to the Equator through Calais in Artois, France, it will be exactly 10

000 kilometres long. This is because, just after the French Revolution, a metre was defined asa ten millionth of the distance between the North Pole and the Equator along a line which

passes through Calais. Nowadays, this isn't considered accurate enough so they use aparticular colour of light and count the number of waves in it instead. This older

measurement varies quite a bit anyway because of things like rocks expanding in summer andcontracting in winter.

If this document is on a piece of paper, that paper will be 0.211 metres wide (211 millimetres

or 21.1 centimetres) and 0.297 metres high (297 millimetres or 29.7 centimetres). Therefore,a metre is about three and a third A4 pieces of paper long. That means that if you started at

the equator with a large stack of A4 sheets of waterproof paper and put them end to end fromthere to the North Pole, you would need 33670033 and two-thirds of them. That would make

a pile 3367 metres high, which is not actually that much if you think about it in terms of shelf


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space in libraries, bookshops and so on.

Many other units in the metric system are defined using the metre. Area is length times depth,

or breadth, or height, and so on. It can be measured in square metres. If this is a piece ofpaper, it has an area of 62667 square millimetres, which can be written as 62667 mm 2. This is

the same as 626.67 cm2 or 0.62667 m2, (square metres). I will come back to the issue of how

many pieces of A4 paper would be needed to cover this planet entirely, because it's not simple.

The metric system is also known as the Systme International, SI for short.

The official SI unit of area is the are, which is a hundred square metres. That would be the

area of a square ten metres on a side, or nearly sixteen hundred sheets of A4. However, the areitself is rarely used as a unit of area and it's much more common to use the hectare, which is a

hundred times bigger. This is the unit of area used to measure things like fields and floorspacein large buildings. A hectare is ten thousand square metres, so a square a hundred metres on a

side would have an area of one hectare. This is very close to the area of Trafalgar Square:

Volume is how big something is. A cube has a volume of the length of one of its edges

multiplied by itself, then multiplied by itself again. A cubic metre can be written as m3, i.e.with a 3.

Once again, the SI unit of volume is not the cubic metre but the litre, although science oftenuses the term cubic decimetre for this. A litre, or cubic decimetre (dm3), is the volume of acube with an edge measuring ten centimetres or one decimetre (dm), a tenth of a metre.

Then there's mass. Mass is the quantity of a substance, which is different than its size. For

instance, a litre of outer space is quite likely not to contain anything at all but is still a litre involume. The SI unit of mass is the kilogramme, which is the mass of a litre of distilled water at

4C, the maximum density of water. The base unit, however, is a thousandth of that thegramme, or gram. A million grammes, rather than being called a megagramme, is referred

to as a tonne.

Weight is not the same as mass, although for masses at rest on the surface of the ocean thedifference between the two concepts would be absolutely minute. Weight is the force on an


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object due to gravity. It is related to mass and when people say weight they are usually

referring to mass by the wrong word. To illustrate the difference, here is a picture of NeilArmstrong:

In this picture, Mr Armstrong has a mass of 77 kilogrammes. Here is a picture of NeilArmstrong on the Moon:


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In this picture, minus his spacesuit, Neil Armstrong has the same mass as he had in the first

picture, but he weighs much less.

Weight is measured in newtons. Actually, weight is not measured in newtons very muchbecause nearly everyone usually uses units of mass to describe weight. A newton is how much

force it takes to accelerate a kilogramme by one metre per second per second. In the first

picture, the entire mass of our planet is pulling Mr Armstrong towards it with a force of about754 newtons, but in the second, he is being pulled towards the centre of the Moon with a forceof 126 newtons because the gravity of the Moon is only 1/6 of ours. However, his mass, which

is expressed in kilogrammes, is the same 77 kilogrammes.

Formulae for volume

The simplest shape to work out the volume of is the cube:

A B

This is easy because its volume is simply the cube of the length of one edge. If you call one

edge AB and assume it's 3 cm long, the volume of this cube is (AB)x(AB)x(AB)=AB3, or in thiscase 3x3x3=27 cm3.

Just slightly more complicated is the cuboid. Cuboids have the same number of faces, edges,

corners and angles as cubes but at least two of those faces are proper rectangles rather thansquares. This is a cuboid:

and this is another one:


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Since cuboids have up to three different lengths of edge, their volume is xyz, with each letter

representing the length of an edge. For instance, this box:

has dimensions of 6x9x10.5 cm, and therefore x=6 cm, y=9 cm and z=10.5 cm, giving it avolume of 6x9x10.5 = 567 cm3, or just over half a litre.

While we're at the simple stage, I just want to mention surface area. Going back to the cube

example, which was 3x3x3 cm, it consists of six square faces with edges three centimetreslong. Since the area of a square is the same as the length of its edge multiplied by itself, in this

case 9 cm2, and it has six faces, it has a total surface area of 9x6, or 54 cm 2.

There are also formulae for other shapes with only straight edges, but these are not so widelyused.

In order to measure the areas and volumes of curved shapes, it sometimes helps to use the

number pi (). is the Greek equivalent of our Latin letter P, and stands for perimeter - thelength of the edge of a shape.

Here is a circle:

The line going all the way across iscalled the diameter. The onefrom the centre to the edge is

the radius which is Latin forspoke. The distance all the

way around the circle is calledthe circumference, which is

just a word for the perimeterof a circle.

If the diameter was a piece of string

and you wrapped it round the edge, you would find that you'd be able to do it just over threetimes, with almost a seventh of the length of the string left over. However, the key word here is

almost. In fact, it's not quite a seventh.


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is sometimes said to be 31

7, but that's 3.142857142857142857... (the three dots indicate

that the pattern 142857 goes on forever). is actually, so far as I can remember,

3.14159265358979323846... - those digits go a lot further than that. The first thousand digitsare 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944

5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095

5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 54930381964428810975 6659334461 2847564823 3786783165 2712019091 4564856692 34603486104543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540

9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 33057270365759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724

8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 19070217986094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082

7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 10507922796892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113

4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 42522308253344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303

5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 17122680661300192787 6611195909 2164201989...

Given the known size of the Universe (a minimum of about 27 billion light years) and the

smallest useful length, the Planck Length, the number is unlikely to be useful with a greateraccuracy than about sixty-one digits, but it never actually ends. It cannot be expressed in

terms of any finite sum of fractions, i.e. it is irrational, like almost all real numbers, almost allof which are unknown for that reason. It's also transcendental not the root of any number,

basically.

comes into many volume and surface area calculations. Let's start with a circle. Here is atape measure:

The diameter of this tape measure is 52 mm, making its radius 26 mm. The formula forcalculating the circumference of a circle is 2r, where r is the radius. To work out thecircumference of the tape measure, we multiply r, 26 mm, by , 3.141592653 and a bit, then


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by two. That makes about 163 mm.

The next stage up is to work out the area of a circle. The formula for this is r2. This is where

priority comes in. Looking at the formula without knowing about it, it could mean eithermultiply by r and then multiply the result by itself or multiply r by itself and then multiply

the result by . Luckily, there is a rule about this, which in this case says that all values should

be raised to the power indicated (squared in this case) before you try anything else. There areways round this, but we are doomed to infix notation, which makes this necessary unlesssome rule for right-to-left or left-to-right calculation is used, so it is normally done this way.

So anyway, the tape measure is 26 mm in radius, so the area of the table it covers is calculated

using that formula as follows: multiply r by itself (26 mm x 26 mm = 676 mm2), then multiplythe result by , making nearly 2124 mm2. Each side of a square of the same area would be

about 46 mm long.

Now consider a cuboid like this box containing a roll of aluminium foil:

This is 370 mm long, square-ended and 37 mm wide., so it has a volume of 560530 mm3,

because the width x=37 mm, height y=37 mm and length (depth) is 370 mm. The formula is

the same as one for the area of a square times its length.

Then there's the cylindrical roll of foil inside:

The principle for working out the volume of this roll is the same you work out the area of the

circle shape, then multiply it by the length of the roll. So assuming it's also 37 mm in diameter,which gives a radius of 18.5 mm, the formula will then be:

r2h

where h is the height, i.e. the length, and is multiplied by the volume of the circle. Therefore

the volume of the roll altogether would be roughly calculated as follows: 3.141592653 x(18.5x18.5)x370= 397828 mm3.


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So in other words, the volume of a cylinder is very easy to calculate.

Now for a cone:

The formula for the volume of this shape is1

3r2h. I have found this less useful than the

others, but would tell you, for instance, how much water a funnel could hold or how much icecream fits in a cone shaped ice cream cone, and so on, and the rough volume of objects like

heaps of sand or mountains.

Finally for the volumes, there is the sphere:

This is of course the Moon. It's 3476 kilometres in diameter and is roughly spherical. The

formula for the volume of a sphere is:


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4

3r3

Therefore, the volume of the Moon is roughly (4 divided by 3) x 3.141592653 x (1738 x 1738 x1738 ) cubic kilometres, or 21990642870 km3.

Now for Earth. The equatorial diameter of this planet is 12756 km, although it is slightlysquashed at the poles and not perfectly circular around the equator either. Putting this intothe formula, you get (4 divided by 3) x 3.141592653 x (6738 x 6738 x 6738) cubic kilometres,

or 1281390881653 km3.

Now back to the important problem of how many sheets of A4 paper are needed to cover thisplanet.

The formula for the surface area of a sphere is 4r2. Given Earth's radius in metres, which is

about 6738000, that formula gets us the result of 570521318634582 m2. Since an A4 piece ofpaper has an area of 0.062667 m2, that means you would need almost 9104015169620089

sheets to cover Earth.

Now a couple of questions for you to answer!

1. What is the volume of a piece of A4 paper, assuming it to be 0.1 mm thick?

2. If a square metre of paper weighs 80 grammes, how dense is that paper given the

answer to question (1)?

3. What is the formula for the surface area of a cylinder?

Units of measurement and volume and surface area formulae

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