## Concentric Circles Area Puzzle

This morning I saw this post from Ed Southall (@solvemymaths):

And thought, that looks an interesting puzzle. I’ll have a little go. I think you should too, before reading any further…

Ok, so this is how I approached it. First I drew a sketch:

I assigned the arbitrary variables r and x to the radii of the larger and smaller circles respectively and used the fact that tangents are perpendicular to right angles, and the symmetry of isosceles triangles, to construct two right angled triangles.

I wrote an expression for the required area in r and x. Used Pythagoras’s Theorem to find an expression for x in terms or r, subbed it in and got the lovely answer of 25pi.

*An interesting little puzzle, did you solve it the same way? I’d love to hear alternative solutions.*

## Area of a semi-circle puzzle

If you have read this blog before, you may have noticed I enjoy a good mathematical puzzle ever now and then. This week I had a bit of time to pass and I thought I’d have a crack at this one from Ed Southall (@solvemymaths):

*Well almost, I didn’t have my phone and I misremebered it slightly, so actually had a good at this:*

As you can see, it’s the same puzzle but scaled by a factor if 1/2.

The first thins I did was label the 4 points. I knew ABD was a right angled triangle, as it is a semi-circle. We are told that ABC and BCD are right angled in the question. The calculate the area we need the radius, which should be easy enough to calculate through Pythagoras’s Theorem and Right Angled Triangle Trigonometry.

I used Pythagoras’s Theorem to calculate AB.

I could then calculate the sine of angle ABC, which I called x. I know that ABC and CBD (which I called y) are complementary, and as such Sin x = Cos y, so Cos y = 1/rt5

This meant I could easily work out the length BD using the fact that Cos y = adjacent/hypotenuse.

This left me with the two short sides of the right angled triangle ABD, so the hypotenuse (the diameter of the semi-circle) was easy to calculate:

From this the radius, then the area, follow easily.

When I came to type this up I realised I’d solved the wrong problem, so for completeness sake, to solve the original problem I can multiply by 4 (as I need to enlarge so lengths gave increased SF 2, and I’m dealing with area) giving a final answer of: 112.5pi (or 225pi/2).

*I enjoyed this puzzle, and did it without a calculator. I think had I used a calculator, it may have lost some of its appeal. I would love to see questions like this appear on the non-calculator GCSE paper. The skills/knowledge needed are circle theorems, Pythagoras’s Theorem, Right Angled Triangle Trigonometry, Surds, Area of a Semi-circle and basic number skills. All of which should be within the grasp of a decent GCSE student. There are, of course, many other solutions, some of which are explored here.*

## Carnival of Mathematics #116

Hello, and welcome to the 116th edition of the Carnival of Mathematics *For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic. This is one of two Maths Carnivals, the other being **Math(s) Teachers at Play**, the current edition can be found here.*

Tradition dictates that we begin this round up with a few facts around the number 116. It’s a fine number, as far as even numbers go. It’s a noncototient, which is pretty cool. There are 116 irreducible polynomials of order 6 over a three element field, it is 38 in base thirtysix. 116 years is the length of the 100 years war and 116 is the record number of wins per season in major league baseball, set by the Cubs in 1906 and equalled by the mariners in 2001. It’s prime factorisation is 2^2 x 29, in roman numerals it is CXVI and it can be represented as the sum of two squares! (4^2 + 10^2)

Now, onto the carnival:

**In the news**

Katie (@stecks) submitted a number of fantastic news articles to this carnival. The first of which is this piece discussing Simon Beck and his geometric snow art. Ed Southall (@solvemymaths) also wrote about geometric art on a similar scale, sharing this about Jim Denevan.

The next article Katie shared was this amazing piece from Alex Bellos (@alexbellos) informing us of the amazingly geeky things Macau have done with their new Magic Square stamps! It almost makes me want to take up stamp collecting!

**Maths Applications**

Grace, over at “My math-y adventures” tells us how she’s put her undergraduate mathematical studies to good use by creating a model to help her with a real life decision.

And Andrea Hawksley writes this phenomenal piece exploring the maths of dancing.

But where there’s good, there’s bad, and here you can see the sort of fake real life contexts that irk Dave (@reflectivemaths) and myself.

**A bit of calculus **

John D Cook as submitted a lovely integration trick which I’d not seen before.

Augustus Van Dusen gives us part four of his review of “Inside interesting integrals” by Paul Nahin.

*Interesting numbers*

3010 tangents have been exploring the infinite with this superb post: “my infinity is bigger than yours”.

Over at the mathematical mystery tour they have shown us how to write pi using primes!

Fawn Nguyen has this great exploration of finding the greatest product.

**Geometric thinking**

Jo (@mathsjem) has written this excellent piece on circle theorems.

@dragon_dodo has produced this cartoon which illustrates Colin’s (@icecolbeveridge) position on radians.

**A bit of fun**

The folks over at futility closet have shared this idea about creating an indoor boomerang!

This week saw the UKMT Senior Maths Challenge, and here is my favourite question.

I spent some time at the children’s playground with my daughter, and found this mathematics there.

Manan (@shahlock) over at maths misery turns infuriating into fun talking about fractions and algebra.

**Behind the mathematician **

Behind the mathematician is a great series from Amir (@workedgechaos) who submitted Martin Noon’s (@letsgetmathing) edition for the carnival! (Mine is available here.)

**Podcast**

This month saw the annual maths jam conference, which brought with it the exciting prospect of a live special of “Wrong, But useful” my favourite maths podcast!

*And that rounds up the 116the edition of the Carnival of Mathematics. I hope you enjoyed reading all these wonderful posts as much as I did. If you have a post for the next carnival submit it here.*

## Circle Theorems: Should we bother?

On the most recent edition of “Wrong, but useful” cohost Colin Beveridge (@icecolbeveridge) had a bit of a rant about circle theorems. He feels they are pretty pointless, and he is in a fairly good position to discuss this, as he spent a decade researching the topology of the sun, basically circular in nature, and never used any of them. He says that he has only ever used one once, and that was to find the centre of a circle (this use is the most practical use if a circle theorem I can think of).

The discussion came about because the other cohost, Dave Gale (@reflectivemaths) was talking about when he trained and the things he hadn’t encountered before. His experience reminded me very much of my own. While I was training to teach I was also working to ensure my subject knowledge was entirely up to scratch, and that I was familiar with the syllabus. There I discovered Circle Theorems, and they were pretty new to me. I don’t know if I’d ever been taught them, I did know that diameters make right angles at the circumference, and that chords make the same angle in the same segment, so I suppose I may have learned then forgotten them. The one in particular that I was certain I’d never met was “Alternate Segment Theorem”, infact it was something that at first confused me and I spent a long time investigating it during my PGCE year before I was completely satisfied that I understood it fully and could teach it.

These Circle Theorems seem to stand alone in the syllabus, they seemingly have no direct link to any other area of the maths GCSE, and they certainly seem to have no real practical use at all and if there is an answer (other then “never”) to the question, “when will we ever use Alternative Segment Theorem in ‘real life’?” I’d love to here it!

**So, should we be bothering with them?**

The usual pro-circle theorem argument goes along these lines: “It’s a nice introduction to mathematical proof which gives students a good grounding for future proofs.” This doesn’t really wash with me. The questions we ask are not very stringent as far as proving goes and more often than not the teaching is focused around “this is how you get the marks on this question,” than any actual proof.

However, I do feel that they have a place on the GCSE syllabus. I used to sit in the anti-circle theorem camp, but my views on this have changed. The more I get into circle theorems the more I love them. And the fact that they don’t have a point just adds to it. The circle theorems are beautiful. They show geometry at its finest, and they have been derived purely because someone wondered about the properties of a circle, not because there was a problem that needed fixing.

Students should be exposed to this kind of maths. They should be allowed to investigate these theorems, and allowed to conjecture about them, before trying to prove them. I think they very definitely should stay in our syllabus, but that we need to address the way they are taught, and assessed. Take the mechanical nature away from the topic and allow the beauty of the maths to prevail.

## Hippocrates’s First Theorem

Over the half term I was doing some reading for my MA and I happened across Hippocrates’s First Theorem. (Not THAT Hippocrates, THIS Hippocrates!)

Here is the mention in the book I was reading (Simmons 1993):

It’s not a theorem I’d ever come across before, and it doesn’t seem to have any real applications, however it is still a nice theorem and it made me wonder why it worked, so I set about trying to prove it.

First I drew a diagram and assigned an arbitrary value to the hypotenuse of triangle A.

I selected 2x, as I figured it would be easier than x later when looking at sectors.

I then decided to work out the area of half of A.

A nice start – splitting A into two smaller right angled isosceles triangles made it nice and easy.

I then considered the area b. And that to find it I’d need to work out the area the book had shaded, I called this C.

Then the area of B was just the area of a semi circle with the area of C subtracted from it:

Which worked out as the area of the triangle (ie half the area of A

)as required.This made me wonder if it worked for all triangles that are inscribed in semi circles this way – ie the areas of the semicircles on the short legs that fall outside the semicircle on the longest side equal the area of the triangle.

My first thought was that for all three vertices to sit on the edge of a semi circle in this was then the triangle must be right angled (via Thales’s Theorem).

I called the length eg (ie the diameter of the large semi circle and the hypotenuse of efg) x and used right angled triangle trigonometry to get expressions for the two shorter sides ef and fg. Then I found the area of the triangle:

Then the semi circles:

I then considered the diagram, to see where to go next:

I could see that the shaded area needed to be found next, and that this was the area left when you subtract the triangle from the semicircle.

I could now subtract this from the two semi circles to see if it did equal the triangle.

Which it did. A lovely theorem that I enjoyed playing around with and proving.

I think there could be a use for this when discussing proof with classes, it’s obviously not on the curriculum, but it could add a nice bit of enrichment.Have you come across the theorem before? Do you like it? Can you see a benefit of using it to enrich the curriculum?Reference:Simmons M, 1993,

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