Every curve in these figures is part of a circle

0:00Here are five drawings. Look at them for a moment before we say anything about them. Two circles side by side, just touching. A ring. Three circles in a row, each overlapping the next. A face, with ears and eyes. And two strokes crossing, with a small circle at each of the lower ends. Now the task. Copy them. By hand, freehand, no instruments. You can do it, and it will look roughly right, and it will not be right.

0:32Because there is something these drawings all have in common that your hand cannot promise. First, a word we need, used in a slightly wider sense than you might expect. A curve here means any stroke a pencil can trace on paper. This wiggle is a curve. This closed blob is a curve. And so is this straight line. A straight line counts. That sounds like cheating, but it is the reason we can talk about a whole drawing at once.

1:05A figure made of straight strokes and rounded strokes is a figure made of curves, and we do not have to keep separating them. Now the question that builds everything else in this video. Here is a point. Call it P. Mark a point that is exactly four centimetres from P. There it is. Four centimetres, straight out to the right. But that was a choice, and there was nothing special about that direction.

1:35Here is another point, also exactly four centimetres from P, on a slant. Same distance. Completely different place. So here is the real question. What would it look like if you marked every single point that is four centimetres from P? Let us just do it and watch. Two points. Then eight. Then forty. Keep going and the marks stop looking like a scattering and start looking like a shape.

2:06And there it is. They close up into a circle. That is not a coincidence and it is not an approximation. That is what a circle is. A circle is every point at one fixed distance from one chosen point. Not a round shape. A rule about distance. Which means the moment you say the point and say the distance, the circle is already decided. You have not drawn it yet, but there is nothing left to choose.

2:39So a circle carries exactly two facts, and they both have names. The point you measured from is the centre. The fixed distance is the radius. Notice where the centre is. It is not on the circle. It is off it entirely. It is the one place in the whole picture that is the same distance from every part of the curve. Give me a centre and a radius and I can draw your circle without seeing it.

3:08Give me your circle and I can find them both. We will do exactly that, shortly. Which brings us to the instrument, and to what it actually does. A compass holds two things a fixed distance apart, and lets them turn. That is all it does. It cannot draw a straight line, it cannot draw a wobble. It holds a distance, and it sweeps. So every mark it can possibly leave is a circle, or a piece of one.

3:39And that is why there is a step before the drawing step. Lay the compass on a ruler. Open it until the tip sits at zero and the pencil sits at four. Now place the tip and sweep. Skip the ruler and you get the right shape at the wrong size, which is not the same as being nearly right. Now the interesting direction. Someone hands you an arc and asks you to copy it.

4:08The rest of its circle was never drawn. All you have is this piece. Where does the compass tip go, and how far open? Watch what happens as we try a centre and swing a test circle through it. Too close, and the test circle curls tighter than the arc. Too far, and it runs flatter. There. One position, one opening, and the test circle lies exactly along the drawn arc.

4:36And here is the useful part: that position is the only one that works. Take any three points on the arc, and exactly one centre is the same distance from all three. So copying an arc is not a knack. It is a problem with an answer. Let us use that. Here is a line from A to B, eight centimetres long, with a point X somewhere on it. Over the first part, from A to X, a half circle bulging upwards.

5:09Over the second part, from X to B, an identical half circle bulging down. Identical. Same size, same shape, opposite way up. So where is X, and how far do we open the compass? If the two halves are identical, they span equal lengths. Eight split into two equal parts puts X at four centimetres. And a half circle spanning four centimetres — that span is the full width across its circle, so the radius is half of it. Two centimetres.

5:44Not measured off the drawing. Read off the line. Now something that catches people out. Take a fixed span, eight centimetres, and draw a half circle over it. Radius four. Now draw a shallower arc over the same span. Flatter. Less bulge. What happens to the radius? Most people expect it to shrink, because the arc looks smaller. It grows. A flatter arc is a piece of a much bigger circle — a circle so wide that the piece you can see barely bends.

6:21Bulge out by one tenth of a centimetre over that same eight, and the radius is over eighty centimetres. And the half circle is the tightest arc there is over a given span. Nothing over eight centimetres can have a radius under four. Which sets up the shape that needs the most care. An eye. It is bounded by two arcs that meet at both ends. One curving up, one curving down.

6:51The temptation is to plant the compass once and swing both. It cannot be done. Look at where a centre has to sit. For an arc bulging upwards, the centre is below the line. For an arc bulging downwards, the centre is above it. Same two endpoints, same opening, two centres — one on each side, the same distance out. Exactly two positions work, never one. So two arcs, two places to stand the tip.

7:24Now go back to the five drawings, because you can read them properly. Two circles touching at a single point: their centres are exactly two radii apart. Any closer and they cross at two points instead of one. The ring: one centre, two radii. Circles sharing a centre can never meet, because a point cannot be two different distances from the same place. The row of three: each overlaps its neighbour, so the centres are less than two radii apart. But the first and the third do not overlap, so they are more than two radii apart.

8:05That is a real constraint on the spacing, not a matter of taste. And the face is a large circle, two smaller ones for ears, two for eyes, a dot for a nose, and one straight stroke. So that is the whole of it, and it is smaller than it looked. Every rounded stroke in every one of those drawings is part of some circle. And every one of them is the same two decisions. Where does the tip stand, and how far is the pencil from it.

8:39Centre and radius. That is the parent of all of it — the arc, the half circle, the ring. You will meet it again as a hole cut in a square, as a square whose sides bulge outwards, and as the curved roof of a house. Every time, the question will look different and be the same question. Next time: building a figure from the measurements you are given, one instruction at a time.