Dear Uncle Colin

I just bought a new set square and noticed it had a couple of extra marks – one at seven degrees and one at 42 degrees. Have you any idea what those are for?

– Don’t Recognise Extra Information Engraved on Calculus Kit

Hi, DREIECK, and thank you for your message!

I didn’t know this, and had to look it up. It turns out, one convention for drawing things in three dimensions uses exactly these angles: lines going “across the way” are 7 degrees above the (negative) $x$-axis; lines going “back the way” are 42 degrees above the (positive) $x$-axis; and vertical lines go directly up. A cube might look like this:

Isn’t that pleasing?

### But why those particular angles?

It’s not just for simplicity and aesthetics, although those are factors. I’m told they make it unlikely for lines to intersect at important points (although I don’t quite see how that’s more true for these angles than any other pair).

My favourite reason, though, is that it’s possible to find these angles quite simply using squared paper: because $\tan(7º) \approx \frac{1}{8}$, you can approximate the slope by moving eight squares to the left and one square up from your chosen point. Similarly, $\tan(42º)\approx \frac{9}{10}$, so moving ten squares across and nine squares up would get you an excellent approximation to the gradient you need.

Hope that helps!

– Uncle Colin

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.