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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. Today we will look at taking our Polygon formula to the limit.

Yesterday’s formula

So, given the number of sides and the length of each one.  We found the formula

Now let’s consider the Circumference

For this post I’m going to add Circumference into the mix. We know that word for circles but we can use it for polygons too – its just another word for perimeter – the distance 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).

What happens when S gets very big?

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

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

This shows that 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. That’s what we get by taking our our Polygon formula to the limit.

This is why I love maths! Everything fits together!

A short trip to Scotland

There is a story about mathematicians that I think I first read when I studying for A-level.


An astronomer, a physicist and a mathematician are on a train in Scotland.  The astronomer looks out of the window, sees a black sheep standing in a field. “All the sheep in Scotland are black!”, comments

“Oh no” says the physicist. “Only some Scottish sheep are black.”

This is the cue for the mathematician to get involved. “No,” he tells his friends. “In Scotland, there is at least one sheep, at least one side of which appears to be black, some of the time.”

I remembered this story yesterday when posting the problem about the dogs, which I reckon wasn’t written by the same mathematician. I’ll explain why in my next post, where I will also give the answer!

How many small dogs? A maths problem from the Independent

Sometimes the time I spend scouring Facebook pays off, and I find a maths problem posted with the a discussion that follows.

This one comes from the Independent’s Website and the story that goes with it says its for a 7 year old,  I’d be interested to see a seven year old tackle it – I have no doubt that a bright one could!  I’m going to share it will my year 10 (15 year old) student this afternoon, and I’ll report back tomorrow, with an answer!

World Cup Time! – Kaliningrad

For the next few posts I’m going to post about Maths and the world cup , and from some of the countries competing. Today its not on one on the countries, but one of the cities where matches are taking place.

Kalilingrad, where England play Belgium is in a part of Russia now separate from the rest of the country but it used to be a German City called Konigsberg.  The city is built on the River Pregel, which flows through the city leaving a number of islands. In the 18th Century there were 7 bridges across the river, like this

The citizens of Konigsberg liked to challenge visitors with the following task – Can you walk around the city, starting and finishing at the same point  and cross each bridge only once. No body was able to do this, but neither could anybody show that it was impossible.

The the Swiss mathematician Euler got involved – and more of him when I get to Switzerland. He showed that by simplifying the map to just dots and lines, it could be shown that it was impossible, and with that started a whole new branch of mathematics called Network Theory, which later became part of the whole new area of Topology.  All because the people of Konigsberg liked to challenge their visitors

What Euler did first was that Euler draw the map as lines and dots, removing the ‘Cityness’ of the map. This is a normal thing to do in ‘mathematical Modelling’ – Strip what he have down to the basics

I won’t use all his mathematical language here but you might see the sense of what he said.  Each point has a number of lines and Euler called this the ‘order’. He showed that to be sure of being able to do the walk, all the points needed an even number of lines. He also showed that you can have two points with an odd number, just so long as you didn’t want to get back to where you started (and started at one of the ‘odd’ number points.

Why not draw some of your own diagrams and check this rule out?

Are GCSE’s getting harder?

That’s a rather open question! And to be honest, I’m not going to give a full analysis here, and certainly not across all subjects!

But I have noticed in recent GCSE exams, more is being asked of the student than just applying the maths they have learnt in the class room.

All exams still have a number of what I call ‘Book Questions’ – ones where the student who has studied their coursework well should be able to answer – Solve an equation, read information from a graph.

There are also ‘Problem solving’ questions – Work out the numbers for a specific situation. These do require planning the work and  sometimes drawing knowledge from different parts of the syllabus. Though more challenging, questions like this have appeared on exams for years.

The more challenging questions I have noticed on recent exams ask for a critical input from the student. These questions reward more than just book learning – Really being at ease with the subject is needed.

This is a question from the higher paper and two things to note – Its question 26 so its from the later part of the exam, and its only worth 2 marks.

This means there shouldn’t be a lot of work involved, but its aimed at the more able student.  Leaving those two things with you, I’ll give a solution tomorrow!





Where is my missing £1

This is a puzzle I’ve known since I was 10, and then yesterday a friend tagged me on it on Facebook. The solution has now been added to the post.  Try answering it first before looking at the spoilers!











OK  –  Here is the answer! 

The skill in solving problems like this is noting down the information you have and trying to ignore any misdirection.  And boy, does the setter of this problem go in for misdirection!

And if you are thinking, that means bringing in skills learnt in an English comprehension class, you’d be right. School subjects may be split between Maths, English, Science, History – but in life things are never so simple. Actually I like to persuade students to write their answers in full English sentences, especially in problem solving cases.

In this situation, we have a situation involving some money, and there is money coming in (the money that each guest pays at the start) and money coming out (to the guests, the hotel and the bell boy).

Each Guest pays £10. That’s £30 ‘in the system’

But who has this money at the end?  The hotel has £25. The bell boy keeps £2. Each of the guests has £1 return to them by the bell boy. That is a total of £30 (£25 + £2 + £1 + £1 + £1)

Or if you prefer lets look and gains and losses. The guests have spent £9 each, making £27.   £25 goes to the hotel, £2 to the bell boy. A total of £27. Gains = Losses. Just as we’d expect.






So lets remind ourselves of the question. This asked us to add the £9s that the guests paid in and £2 the bell boy kept.  But these numbers are from different parts of the question! (As highlighted in grey and blue)   There is really no reason why they should be added. The fact that the total came close to £30 was a coincidence (or not, since the questioner chose the numbers to confuse!)

So in summary, ignore the last paragraph until you have come to your own understanding about the ins and outs. And then you won’t fall into the trap of adding an ‘in’ amount to an ‘out’ amount.