Degrees
Cut the full turn into equal pieces, and see how many degrees there are in each.
- Number of pieces
- 8
- The radius
- 4 cm
- All the way round
- 360°
- The arc of one piece
- 3,1 cm
- One piece
- 45°
Regnestykket
All the way round ÷ Number of pieces
360° ÷ 8 = 45°
Card: 360 ÷ n
Why exactly 360
There's nothing in a circle that says 360. It is an agreement, and a very old one — it comes from Babylon, where people counted in sixties and twelves instead of tens. The agreement has lasted because the number is easy to divide: 360 can be divided by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more without a remainder. Try the slider and watch when the number comes out even and when a decimal appears. Most settings come out even, and that is the whole reason that particular number was chosen.
The pieces are the same, and there are n of them
Unfold, and look at the row. All the pieces face the same way, they have the same arc and the same corner down at the tip. None of them is bigger than the others. When together they make exactly one full turn, and each has the same number of degrees, then each piece must have one full turn divided by the number of pieces. It isn't a formula you have to remember — it is the only answer that can fit, once you have seen that the pieces are the same.
Notice what the radius does NOT do
Drag the second slider and watch the figure: nothing happens. The drawing is always scaled to fill the screen, so a circle with radius 2 and one with radius 6 look exactly the same here. Look at the numbers underneath instead. The arc of one piece changes every time you drag — but the number of degrees stands completely still. That is the most important discovery on the page. An angle says something about how much something turns, not about how big it is.
The arc is a length, the angle is a turn
In the numbers underneath, the arc of one piece comes just before the number of degrees. The arc is measured in centimetres and gets longer when the circle gets bigger. The number of degrees is measured in degrees and only changes when you divide into more or fewer pieces. They are two completely different questions about the same piece: how far is it round along the edge, and how much do you turn. Two corners can have the same number of degrees and be very different in size. Don't mix them up — that is exactly where most mistakes with angles come from.
The half, the quarter and the right angle
Set the slider to 2. Now there are two pieces, each of 180 degrees, and each of them is half a turn — a straight line. Set it to 4: four pieces of 90 degrees, and 90 degrees is exactly the corner called a right angle. That is the corner all angles are measured against, and it is the only place in the house where two walls meet properly. Set it to 3, and you get 120 degrees, which is the corner in the hexagon bees build with.
Where you'll meet degrees again
An angle grinder and a mitre saw have a scale in degrees, and a 45-degree cut is what you need for two boards to meet in a clean corner. A roof has a pitch given in degrees. A pipe that has to go round something is bent at particular degrees, and the bends you can buy are exactly the numbers that divide into 360. A lorry wheel turning, a crane swinging, a window opening — all measured on the same dial, with the same old agreement that all the way round is 360.