What is a Regular Quadrilateral?

Have you missed me? It's been over a whole glorious month since I last posted. I've been lying on my sofa reading nineties teen romance novels and leaving the house only to visit Ikea for free tea and to pretend I live there. 

But anyway, I'm back and have I got a mathematical ponderance for you!

First, grab a piece of paper and a pen (or open a notepad file, for the more evolved among you) and write down (type) the definition of regular (as in, a regular polygon). 

Done? OK. Hands up who wrote this:

"All the sides are the same length".

And hands up who wrote this:

"All the sides are the same length and all the angles are equal".

Did any of you just write this:

"All the angles are equal".?

A more important question, perhaps, for teachers, is this: what do you tell your students?

Now another little exercise for you. Is this statement always, sometimes, or never true:

"If a polygon has all equal sides then all of the angles must be the same size".

Let's think for a moment. We know it is true for triangles, although you might not be able to prove it, or even justify it, beyond the fact that your Year Five teacher told you it was true during a particularly soul-crushing numeracy hour. We have also always assumed it was true for "big polygons" like octagons, decagons, etc. But is it true?

There is one type of polygon for which the statement is definitely not true. Quadrilaterals. Sorry if the title spoiled this major reveal for you. A square is equilateral and equiangular. However, a rhombus is equilateral but the angles are not all the same size. 

OK, here's another one for you:

"If a polygon has all equal angles, then all of its sides must be the same length". Always, sometimes or never true?

Again, think about triangles first, and then think about hexagons etc. Can you prove your conjectures?

Again, the statement is clearly false for quadrilaterals. A rectangle is equiangular but not equilateral. 

So what is a regular quadrilateral? Is it a square, a rhombus, or a rectangle?

My initial thought (because I admit, I didn't actually know the correct answer), was that a rectangle is regular. I thought this because geometry mostly comes from Greece, and in Greece, they're mostly bothered about angles. Hence the word polygon: "poly" meaning many and "agon" meaning angles. So a hexagon is literally a shape with six angles. In the UK, we're more likely to say a hexagon is a shape with six sides. So I thought a regular polygon would mean a shape with regular angles. 

However a quick tussle with my favourite search engine revealed that a regular polygon must be both equilateral and equiangular. 

I'm ashamed to admit I think I might have taught students that regular just means equilateral. Or, even worse, I think I might have even implied that equilateral shapes were always equiangular! It's funny how such a simple little definition can be messed up because you think you understand it perfectly (after all, I was taught it in year five). Maths teachers like me need to make sure we are completely clear about these things. Maths doesn't leave much room for error, and these definitions need to be water-tight. 

On that note, have a good first day at school!

Emma x x x 

PS I am of course talking about convex regular polygons. Non-convex (star) polygons are of course a whole other kettle of fish!