The total interior angle of a triangle and other polygons

The best answer is not 180° despite what everybody says! ;-)

This little bit of fun is mostly intended for “young mathematicians”.

Equipment you’ll need

You’ll either need some chalk if you are working on some ground you can draw on (such as a playground), or you’ll need something like rope or string, and then some weights for the corners of your shapes.

PART A

(1) Stand on a single spot in a space with lots of empty space around you, such as a garden

(2) Stick your arms straight out in front of you and lock your hands together to make an “arm pointer”

(3) Point your “arm pointer” at something (choose something that is easy to point at such as a particular garden ornament)

(4) Standing on your spot, rotate all the way round so that your arm pointer comes back to pointing at the same thing it was pointing at when you started your rotation.

What you just did in (4) is called “a whole rotation” or “a whole turn”.

PART B

(1) Draw a big triangle on the ground. It needs to be big enough to walk around. As an alternative to drawing your triangle, you can stretch out a length of rope or a long piece of string so that it is taught and straight along the three sides of your triangle and then place something heavy enough to keep the rope or string in place in each of the three corners of your triangle.

(2) Start by standing on one corner of the triangle.

(3) Stick your arms out straight in front of you with your fists locked together so that your arms become like “one arm”. Your arms should be going straight forwards from your shoulders. When you turn, turn your whole body along with your arms. Let’s call this your “arm pointer”.

(4) Start with your “arm pointer” pointing down one side of the triangle (still standing on the first corner)

(5) Turn your arm pointer (and your whole body with it) so that it points down the other side of the triangle (still standing on the first corner).

(6) Now walk forwards along the side of the triangle that you just turned towards. Do not rotate while doing this.

(7) Whichever direction you turned in on the first corner, throughout what follows keep turning in the same direction. This is very important.

(8) When you get to the second corner of the triangle, your arm pointer is still pointing in the direction of the first side you walked along, but it is now sticking out of the end of the triangle.

(9) Looking back over your shoulder but without turning, you should now be able to see the triangle behind you.

(10) Turn your back through the angle of the second corner (that angle will be “behind you”).

(11) You should now have your back facing in the direction of the second side of the triangle.

(12) Walk backwards along the second side of the triangle until you reach the third corner.

(13) You should now have the triangle in front of you again, and your arm pointer should be pointing in the direction of the second side (because you haven’t turned yet).

(14) Now (finally) turn your arm pointer through the angle of the third corner, and you should be pointing down the third side of the triangle towards the corner where you first started.

(15) If you did the procedure correctly you have now turned (turning in the same direction at each corner) through each of the three angles of your triangle.

The question is, compared to the “whole turn” that you turned through in PART A, how much of a whole turn have you turned through in PART B ?

If you did the procedure correctly you will have just discovered for yourself something that mathematicians call “the total internal angle of a triangle”. The “total internal angle” just means you add the three angles in the corners of the triangle up together.

It is very important to actually DO the procedure. Not so good, for example, to merely imagine yourself doing it.

Don’t read this next bit until you have actually done the procedure

What you hopefully may have noticed in going through the procedure is you started with your arms pointing down one side of the triangle, and finished (having turned in the same direction through each of the three angles of the triangle) pointing in the opposite direction along that same side. You were facing up that side, you are now facing down that side.

In other words you have, in total, turned through half of a whole turn. So the total interior angle of a triangle is exactly this: it is half of a whole turn. What’s more, this “half of a turn” is not arbitrary. There has not been any human choosing involved with coming up with this half of turn. There is no avoiding it. Three sides … half a turn.

So what’s wrong with calling this 180°? Nothing at all as long as you appreciate that this is arbitrary. Once you have decided, as the Babylonians apparently did first, to split a whole turn into 360 tiny bits of turn, half of 360 certainly is 180. In addition to which 360 is a lovely number which can be factorized in a smorgasbord of ways, making it ideal for laying out cities or putting up buildings, or designing sun dials, or henges. Even so be clear that a human choice has been made, whereas with “half of a whole turn” it has not.

You could if you wanted, instead of saying a whole turn equals 360, say a whole turn equals 100 (in which case half a turn would be 50) or say a whole turn equals 12 or even say a whole turn equals 6.28 , in which case half a turn would be 3.14 … and you never know where one or another of these choices might turn out to be helpful.

Extra fun for those who want it

You can repeat the procedure above for a 4, 5, 6, 7, etc. sided shapes, although you may quite probably discover that as you get much beyond 6 or 7 sides it becomes increasingly difficult to draw them with chalk on the ground.

What you will discover if you do this, is that for each additional side you add to your shape, the amount of turn that rotate through in turning through all the interior angles is an extra half of a turn for each additional side. So a 3-sided triangle makes you turn half of whole turn, a 4-sided “quadrilateral” makes you turn through a whole turn, a 5-sided shape (such as a pentagon) makes you turn through one and half turns, and a 6-sided shape (such as a hexagon) makes you turn through 2 whole turns. That is to say, an additional half of a whole turn for each additional side added.

Can you think why this could be?

Take care!

:-)