📐 The Pi Garden
Angles always add up
An angle is an amount of turn. Once you see that, every rule about angles comes from the same idea.
Draw the same angle twice, once with short arms and once with long arms.
Most people guess the one with long arms is bigger.
But the length of the arms makes no difference.
An angle only measures turn.
Start here
An angle is a swing
Picture a hinge with one arm pinned down and the other free to swing. The angle is how far the free arm has turned away from the fixed one.
Your turn. Drag the brass arm round the hinge. Watch the number go up, and look out for the turns that have their own names.
40°
acute: less than a quarter of a full turn.
The shaded wedge is the angle, the amount of turn between the arms. It doesn’t depend on how long the arms are.
So if an angle only measures turn…
then the size of the drawing makes no difference.
The trap
Which angle is bigger?
There are two angles below. One is drawn small and the other takes up half the panel. Which corner has the bigger angle? Pick your answer first and stick with it. Then drag the small angle onto the big one, corner to corner, to check.
Two clocks, one time: a wristwatch and a big wall clock both show quarter past six. Move the slider to change the time on both.
The wall clock’s hands are far longer, but at the centre both pairs of hands have turned the same amount, so the angle between them is the same.
A straight line is half a full turn.
That’s always 180°.
Why, part one
Split a straight line into angles
You’ve seen on the hinge that a straight line is half a full turn, 180°. What happens if you split it with another line? Before you drag, guess what the two numbers below will always add up to. Then try your hardest to prove yourself wrong.
However you cut it, the pieces still make up the same straight line, half a full turn. They add up to 180° no matter how many cuts you make.
Now let’s try the famous one.
It works for every triangle you could draw.
Why, part two
The angles in any triangle
Drag the corners anywhere you like to make the triangle long and thin, squat and wide, or as lopsided as you can. The three angles change, but their total stays the same. When you’re convinced, tear the corners off to see why.
It comes from the last section. If you tear the three corners off any triangle, they fit together exactly along a straight line. A straight line is half a full turn, which is 180°, so the three corners add up to 180° too.
Try it on paper. Draw a triangle, tear the three corners off and push them together point to point. You’ll get a straight edge every time.
If you tear the three corners off any triangle…
they fit together into one straight line.
So the three angles make half a full turn.
Everything here comes from one idea: a straight line is 180°. The three angles of a triangle fit together into one straight line, so they add up to 180°. Four right angles around a point make two straight lines, so they add up to two lots of 180°, which is 360°, a full turn.
Doing the sums
Finding the missing angle
This is how it comes up in exams: a triangle with two angles given and one missing. You know the three angles add up to exactly 180°, so the missing one is whatever’s left over. Pick a pair and compare the tempting wrong answers with the right one.
Think of it as pocket money. Every triangle has exactly 180° to spend on its three corners. You know what two corners have spent, so the third gets whatever’s left. Work out 180 minus the two you know.
You’ll start to see angles everywhere…
once you know they’re just turns.
There’s one between the hands of a clock, at the top of a roof and at the point of a slice of pizza.
In every one, the length of the lines makes no difference.
Pass it on
A clock face is a full 360° (six sixties, if you’ve done the sixty lesson), and the hands make a new angle every minute. The two sloping sides of a roof meet at an angle at the top. Cut a big pizza and a small one into eight slices each, and the point of every slice is the same 45°, however big the pizza is.
So next time someone says the big sprawling drawing has the bigger angle, ask them how far each one has turned. If you can explain why quarter past six makes the same angle on a wristwatch as on a big wall clock, you understand angles properly.
Drag everything you like. The totals always stay the same.