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Taking our Polygon formula to the limit

In yesterday’s post I considered how we can find the area of a many sided regular polygon given the number of sides and the length of each one.  We found the formula

For this post I’m going to add Circumference into the mix; all the way round the shape, the length L, S times

C = L x S

In the formula, we are now going to replace the L with a C, since the more sides we have, L is going to get small, and C is an easier thing to measure.

L = C/S is just a rearrangement of the formula above which we can use to replace L in our area formula

This gives a formula with C in of

A = C2/4STan(180/S)

We now have moved all mention of S to the bottom of our formula.

Now if C is kept the same but S gets bigger and bigger, what does that mean for our formula?

As S gets bigger. 180/S will get closer to 0, and so too will Tan(180/S), S will obviously get bigger, so what does that mean for  S x Tan(180/S).

Now this is where I am going to cheat a little; It may be possible using mathematical techniques to see what happens as S gets bigger…  but I am just going to plug some numbers in

When S = 10,  S x Tan(180/S) = 3.249
When S = 50, S x Tan(180/S) = 3.146
When S = 100, S x Tan(180/S) = 3.143
When S = 1000, S x Tan(180/S) = 3.142

S has to get quite big before the pattern is clear, but it seems that
S x Tan(180/S) is getting close to a very familiar number, π.

So it seems for very large values of S

A = C2/4π

Remember that C = 2πr2 so C2 = 4π2r2

So, A = 4π2r2/4π

The 4π on the bottom cancels with elements on the top and we are left with

A = πr2

Which is the familiar area of a circle! And if you think about polygons with many, many sides you will see they are very close to being circles.

This is why I love maths! Everything fits together!

Area of a polygon

I meant to post this last week, but I could see after my first draft I’d made a mistake in the Maths!  Yes it happens……

Anyway, many of my posts have addressed specific exam questions but sometimes I just get a bug about something .. and then I think,  why have I never thought about that before!!

I don’t know of any maths course this is on, but hey, I just wanted to know, as I wrote my last two posts….   Is there a formula for the area of a regular Polygon!

If it is 3 sided or 4 sided – a triangle and a square – then we know the formula for area, but I was thinking – what about a formula that works for any regular polygon – That is to say, one with all the sides the same.

 

 

 

 

Here is a polygon and lets say the length of all the sides is L.  You can count the sides here and see there are 8 – this is an octagon  – but let this represent ‘any polygon’ with a number of sides S.

What I will be looking for is a formula where A = something with L and S in, as they are the two ‘properties’ of the polygon that might change in our ‘general’ polygon

The art of finding the area of any unfamiliar shape is to divide it into shapes for which you know how to find the area.  Any polygon can be divided into triangles by drawing lives from each corner into the middle.

How many triangles?  S  – One for each side of the polygon.

 

What is the area of each of these triangles?  For this we need to know the base and the height; The base is L. To find the height we will need to use some trigonometry. Half of the triangle is a right angled triangle. I am going to use the angle at the top of the half triangle, which I have labelled x

The full angle at the top is 1/S of the 360 degrees at the centre  =

360/S.   x is 1/2 of this  = 180/S

Tan(x) = L/2  /  h  (Opposite/Adjacent where the ‘opposite’ is half the base  and h is what we will call the height, for now.

Add in some detail and rearrange

Tan(180/S) = L/2h
Rearrange again to give h = L/2Tan(180/S).

Area of the small triangle is (using 1/2 x b x h)   = L2/4Tan(180/S).

This is only 1 triangle out of S triangles, so the formula for the whole polygon is

Now, as I said at the start, I worked out this formula a week ago, bit I wanted to check my work because it looked a bit cumbersome….. But it does seem to stand up, and I’ll show you how.

There are two polygons for which we know another formula; The Triangle and the square (S = 3 and S = 4).  Lets see what happens if we make S= 4

A =  L2 x 4 / 4Tan(180/4)

Tan(180/4) = Tan(45) = 1 – so we cand leave this term out. Also the 4 and 4 can cancel, and we get

A =  L2 – the simple and familiar formula for the area of a square!

Checking this for the triangle is more tricky because we to find the ‘normal’ area for a regular 3-polygon – or as we usually call it, an equilateral triangle.  For this we need our old friend, Pythagoras’ Theorem.

Using the theorem in half the triangle, we get

(L/2)2 + H2 = L2

H2 =  L2 – L2/4  = 3L2/4

H = √3L/2

so A = √3L2/4

Now lets see what we get from our polygon formula with S=3.

Tan(180/3) = Tan 60 = √3

A = (L2 x 3)/(4 x √3)

Remember that 3/√3 = √3

so A = √3L2/4

The same as with the direct method!

That, I claim, justifies my formula.  If anyone can think of an alternative way of finding the area of a pentagon or hexagon, then the formula can be checked for these shapes too.

Now, interesting things happen to our formula if S gets bigger and bigger, but that will have to wait for another post!

Just what is an external angle?

In yesterday’s post I wrote about how we can find the internal angle in regular polygons, and how the total of all the internal angles have the pattern 180, 360, 540, 720  ….    This patterns isn’t too hard to remember and most of my students remember this as a pattern

Actually its easier to understand what is going on if we look at the external angles.

It’s important to know what the external angle is.  It is NOT the angle all the way round the outside

It is the angle between each line.. if it were drawn longer..  and the nest line round the shape in that direction.

It is useful to imagine you are walking around the shape. The external angle is the angle you  turn your body round at each corner.  Then once you have walked round the whole shape..  and ready to start again ..  you have turned the full 360°

So the size of each external angle = 360°/Number of angles.

The number of angles is the same as the number of sides, of course.

That is an easier formula than the one we saw for the internal angles, but I always get curious in these situations. We have two formulas…  do they work together?

For a given regular polygon lets say it has S sides (so also S corners, A internal angles and S external angles). Let A be the size* of each internal angle and X be the size* of each external angle.

*They will all be the same because this is a regular polygon. This sudden move into the language of algebra is because we don’t know how many sides our polygon has – we are looking at all polygons at the same time.

X = 360/S (Today’s formula)
A = (S-2) x 180/S (Yesterday’s formula for the size of each angle)

Also, X = 180 – A : The two angles make a straight line. Look at the diagram.

To show these formulas all say the same thing, we need to combine two of them and show we get the other one. This can be done a number of ways, but I’ll only show one here.

Take X = 180 – A  and substitute in the formula for A

X = 180 –  180(S-2)/S

X = (180S – 180(S-2))/S – I’ve made the whole equation ‘over S’ by including the first 180 in the fraction)

X = (180S – 180S + 360)/S  I’ve multiplied out the bracket on the top

X = 360/S – Because 180S – 180S = 0!
And so we get to the other formula for X

How many angles can you fit on one post?

A regular polygon is any shape where all the sides are

the same length. Some of these shapes we might not immediately recognise as a ‘regular polygon’ because we have another name for the shape – Square and Triangle – specifically an ‘equilateral triangle’

 

In this post we are going to look at how big each angle is in each shape.

 

 

We know that angles in a triangle add up to 180° – That is something we learn – and that a square is made up of four right angles – 4 x 90° = 360°

The next polygons – the pentagon, hexagon,  heptagon and octagon.

The hexagon is probably the most familiar of these words, the pentagon is perhaps most famous for the US department of defence. The other two words might be new to you (though where I was at university the Octagon was the main conference building – it was where I graduated!)

The sums of the angles in these shapes are 540°, 720°, 900° and 1080°.

You might see a pattern there – each number goes up by 180°.

We can turn this into a formula where S is the number of sides of the polygon

Total of Angles = (S – 2 ) x 180.

From this we can work out the size of each angle in a regular polygon. Because the sides are equal, so are all the angles.

Each angle is 180(S-2)/S.

In this post I have been looking at the ‘Internal Angles’.  Actually, its easier to show that these formulas work if we consider the external angles.  What are they, you might wonder?

Well that will be the subject of the next post!

How to construct a ….

When I was teaching at the local FE college, there was one part of the GCSE where I sort of cheated,,,  I let YouTube do the work for me!

This was the set of skills called ‘Constructions’ – which can be hard to recreate on a whiteboard, especially when was one my ‘Motor Skills’…

The premise of these skills is that the student needs to

  • Construct a perpendicular bisector of a line
  • Construct a perpendicular to a line from any given point
  • Construct a line that bisects an angle

The extra catch is these have to be done using a pair of compasses and a straight edge only

There is something a little ‘old fashioned’ about these skills, but they remain on the GCSE Maths syllabus. I’ll also add that I haven’t seen them as much on recent exams papers – Maybe once at most across all three appears for any given season…

For all that they are rather fun to do and understand.

They are hard to do in front of a class; and on a blog too!   So let me give you some links to show how to do them.

 

Angling for some Fun – Part two

In the last post I started to describe how patterns in angles can help us find the size of missing angles without doing any measuring. In this post I am continuing with that theme. First I will show what we mean by Corresponding and Alternate angles.

Then I will show how we can put all this together to answer a question.

For both of these situation, we need one line crossing two others which are parallel.  If we don’t know the two lines crossed are parallel, we can’t use these rules, got it?

And you know how to tell if two lines are parallel? The two lines are never going to meet, however far we extend them.

So, the two angles marked in red – they are going to be equal.  So if we are told the lower one is 50˚  then the higher one is also 50˚.

These is what are called ‘Corresponding angles’

This is like the rule in the last post where we know two angles must be the same if they are ‘opposite’. Angles are also the same if the are ‘corresponding’

Now look at this diagram.

The two red angles are ‘corresponding’ and the green angle is ‘opposite’ one of the red angles.

So all three angles will be the same.

We say the bottom Red angle and the green angle are ‘Alternate’.  ‘Alternate’ is like a combination of the ‘Corresponding’ rule and the ‘Opposite’ rule.

Some people think of this as the Z rule – because the angles in a Z are the same.

I’ve been looking for an exam question that uses all of these but they are hard to find, and I only want one for this post, which would be too long otherwise…  so…  let’s have a look at this

 

A few points about the wording and notation here
1. I across two lines shows those lines are of equal length : AB = AC
2. > on two lines shows they are parallel. AP is parallel to CB
3. ‘Diagram Not Drawn to scale’ means take the information given as true. Don’t check them with your ruler or protractor.

ABC is an isosceles triangle, so <ACB = <CAB = 70˚.  < ABC = 180 – 2 x 70 = 40˚ – because of angles in Triangle ABC add to 180˚

<BAP = <ABC because they are Alternate – (to see the Z shape turn it round a bit)

So x = 40.

To Infinity and Beyond

I’ve been posting a lot recently on answers to GCSE questions – mainly because that is what I have been doing with students –  and I do have ten more days of that.

For today something different a look at something quite unusual…  As the title suggests – INFINITY.

Today’s post is based on some of the work of German mathematician Georg Cantor, a man not often understand by his contemporaries – In fact one said he was ‘aha ed of his time by one hundred years.

What Cantor was trying to do was explore what infinity means, and his revolutionary idea was that there is more than one infinity!  In fact…  there are an infinite number of infinities!

I’m not planning to show where they all come from in this post – I’ll just show that there is more than one.

 

 

If you go out with six friends this week, how will you know there are six of them? Well, OK, that seems like a simple question – because you can count, right?. Yes, but understanding how we can count is the starting point of understanding where all the extra infinities come in. You know you have six friends with you because you have given each a number, starting with 1..  and you got up to six! OK so may not have done that explicitly, but that is what you did.

And that is what we are going to use in a moment to demonstrate infinities.

Two quantities are the same if you can match up one item from one ‘list’ with one ‘item’ of the other… and have none left over. So the numbers 1 to 6 can be matched with your 6 friends with no numbers left over.

Now, this is where we come to some surprising ‘facts’…..  Ask,  are there more whole numbers than ‘even’ whole numbers.  You think, yes? Well look at this…..

 

 

 

 

Even numbers can be paired up with whole numbers with no numbers from each group left out.  This is one of the surprising things we learn about infinity. The number of even numbers is the number all numbers.

Now imagine you are on a ‘chessboard’ floor that stretches for infinity in all directions.

How many squares are there. Infinity – or more than infinity. This, after all ‘infinity squared 

 

 

 

Let’s try counting the squares in the same way that we counted your friends.  Start with one square – it could be the one you are standing on. Thats 1.  Now count the 8 squares around you – 2,3,4… 8

Then the squares around that.

 You can keep going like that for ever  – but that’s OK, we can keep counting for ever.

So even on this infinite board, the number of squares is the same as the number of numbers – Infinity.

 

Now, at this point, if I didn’t know what was coming next, I might be starting to doubt my earlier remark – that there is more than one sort of infinity.

Lets consider now, decimal numbers, by which I mean numbers of the form

1.5434677654433345656

I have 20 digits after the pint there. Lets say we mean numbers that DO end, but we will not specify how many digits they have before they do.

So lets start with the number 1.5

Where do we go to next to count; 1.6  or 1.55? Or 1.5000000001.

There are actually an infinite number of next steps, and thats before we get off our first ‘square’.

And THAT is where the next order of Infinity comes in. Its not possible to ‘count’ the decimals  with the numbers 1, 2, 3 …….  even they go on for ‘infinity’.

There are more than infinity decimals!

Crazy Curves

One last post before I move on from ‘continuous’ curves….

I was going to include this in the last post, but that was already too long.

We have looked at curves which are continuous everywhere, and some which are not – but are continuous for most of the way.

Is it possible for a curve to be discontinuous everywhere?  In theory yes, though we need to consider rational and irrational numbers.

A rational number is any that can be written down, accurately, with numbers.  This includes numbers there are ‘recurring’ like 0.3333333 because this can be written as 1/3, and be accurate

Pi is an example of an irrational number

So if we say that y = f(x) where f(x) = 1 when x is rational and f(x) = 1/x where x is irrational…  then that would define a curve, but one that is so chopped up is would be continuous in only very small sections between rational numbers

Let’s have a look at curves

I’ve been looking at exam questions a lot recently in this diary – well it is coming up to exam time!

In June, the focus is going to be on famous Mathematicians, Over the summer I’m looking to write some entries on mathematical puzzles – If you know some good ones please send them in!

Today though, I’m looking at curves – the lines we can draw to represent certain mathematical equations.

In particular, today and tomorrow I am writing about curves that are ‘continuous’ and curves which are not, which we call ‘discontinuous’. Also, what that means for how we can use these curves to answer questions – That bit is for tomorrow!

What do I mean by continuous? Basically this means that an ant can follow the curve and get to all parts of the curve, without flying off the page or leaving the line. There is a Mathematically precise way of defining ‘continuous’ but I think my ant gives you the idea.

Lets look at some examples. (For this I am going to borrow screenprints from my favourite websites, the link on the Links page)

y = x2

You can imagine the any being able to walk around this line. In fact any line of the form
Axa + Bxb + Cxc+ ….   will be continuous so long as a, b, c etc are positive integers.

 

 

 

 

y = sin (x)

 

And again the ant can walk up and down these curves.

 

Since the line for y = cos x looks similar, we can say that is continuous too.

Lets have a look at some Non-Continuous curves now, The easiest to show is  y = 1/x

This curve is in two parts. Our ant isn’t going to get from one part of the curve to the other without a jump.

 

 

 

 

 

Going back to our trigonometry, Tan (x)  – unlike Sine and cosine – is discontinuous. There are many and regular breaks in the line.

 

 

In tomorrow’s post I will look at some more eccentric examples and show why its important to understand if a curve is continuous or not.

 

 

 

Completing the powers.

Yesterday’s post built up some of the things we need to answer a question where the power is negative and a fraction.

One key point is that for all numbers n

n0 = 1  and n1 = n.

With negative powers we still need to maintain the rule of adding powers.

2x x 2-x  = 2x-x  (and since x – x x = 0)  = 20 = 1

Re-arrange this and we get  2-x = 1/2x

And there the first new rule – a negative power is 1/the positive power

32 = 9 so 3-2 = 1/9

To get the second rule we need to consider how powers can be combined.

(n2)3 = n6  – When you raise a power to a power – multiply the powers

[Consider  n x  n     x     n x n     x    n x n]

 

Now look at

(n2)1/2  =  n1  = n

So what does raise to power of half mean if this involves getting from n2 to n?   Taking the square root!  We could replace 2 and 1/2 with 3 and 1/3 – so see n1/3 means take the cube root – and so on.

n1/x means take the xth root of n.

Just before we get back to the question given, lets just complete the patterns by considering what x3/2 means. I have seen some exam questions that do pose questions like this.

I suggest you split the 3/2  into  1/2 x 3  or,  n3/2 = (n1/2)3

so 43/2  =  (sqrt(4)3)  = 23 = 8. It is generally easier to take the ‘root part first. In a non calculator paper you’ll only be asked about roots you know.

Let’s get back, at last, to the original question.

64-1/2–   Take the 1/2 part first, that means square root, and the square root of 64 is 8.  The – part means take the reciprocal  1/8

What Carol did was take 1/2 of 64, not the square root of 64, so your answer should include a sentence.  ‘Carol didn’t know that a power of 1/2 means square root, not multiply by 1/2’ – then give the correct answer of 1/8.

 

I’ve made this into 2 blogs posts with lots of background but if you can remember the rules given in these posts, this question need not take you long in an exam,