The Angle in a Semicircle

Here is a small magic trick you can do with nothing but a circle. Draw a circle and rule a straight line right across it through the very middle — a diameter. Now put your pencil on any spot you like on the circle's edge and join it to the two ends of that line. Look at the corner you just made at your pencil point.

It is a perfect right angle — exactly 90^\circ. Move your pencil to a completely different spot on the circle and try again. Right angle. Again, high up near the top. Still a right angle. Every single time. The corner is square no matter where on the circle you stand it — as long as the far side goes straight through the centre.

That is genuinely useful. It means a circle is a machine for drawing perfect right angles. No set-square, no protractor: draw a circle, draw a diameter, pick any edge point, and you have a guaranteed 90^\circ. Builders and carpenters have leaned on circle tricks exactly like this for thousands of years to check that a corner is truly square.

The rule, and why it's true

Take any circle and draw a diameter — a straight line right through the centre, with both ends on the circle. Pick any other point on the circle and join it to the two ends of the diameter. The angle you make there is always a right angle:

\angle APB = 90^\circ

It is the angle at the centre theorem in a special case: the diameter is a straight line, so the angle at the centre is 180^\circ, and the angle on the circle is half of that. Half of 180^\circ is 90^\circ — and there is your right angle.

If AB is a diameter of a circle:

Why it works

No measuring needed — the diameter does all the work. Step through the reason.

Wherever you slide P around the circle, the diameter keeps the centre angle at a straight 180^\circ, so the angle at P stays at 90^\circ.

See it for yourself

Drag the slider to walk the point P right around the top of the circle. The triangle stretches, leans, and changes shape completely — tall and thin, short and wide — but watch the little square at P. It never budges from 90^\circ.

Using the rule

The trick in a problem is always the same: spot the diameter. The moment you see a triangle whose longest side is a diameter of the circle, you can stamp a 90^\circ on the far corner — for free, without measuring anything.

Example 1 — find the other angles

A triangle APB is drawn on a diameter AB, and the angle at A is 35^\circ. Find the other two angles.

Because AB is a diameter, the angle at P is a right angle straight away:

\angle P = 90^\circ

The three angles of any triangle add to 180^\circ, so the angle at B is whatever is left over:

\angle B = 180^\circ - 90^\circ - 35^\circ = 55^\circ

Example 2 — bring in Pythagoras

Now you have a right-angled triangle, so Pythagoras' theorem is yours to use as well. Suppose the two shorter sides of the triangle are 6\text{ cm} and 8\text{ cm}. The diameter is the hypotenuse (it sits opposite the right angle at P), so:

AB = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}

And since the diameter is 10\text{ cm}, the circle's radius is half of that — 5\text{ cm}. One right angle unlocked the whole figure.

Example 3 — an isosceles surprise

If the two shorter sides happen to be equal, the two non-right angles are equal too, so each must be (180^\circ - 90^\circ) \div 2 = 45^\circ. A right-angled triangle on a diameter with equal legs is exactly a 45^\circ–45^\circ–90^\circ triangle.

The whole trick rests on one word: diameter. The far side of your triangle must be a line that passes right through the centre of the circle. Only then is the corner 90^\circ.

A chord that stops short of the centre — any old straight line between two points on the circle that doesn't go through the middle — does not give a right angle. The corner it makes could be any size at all. So before you write 90^\circ, always check: does the line go through the centre? If the centre isn't marked, don't assume it. A diameter is the longest chord you can draw, and it always splits the circle into two equal halves — two semicircles. If the line doesn't do that, the 90^\circ rule simply doesn't apply.

This fact has a name — Thales' theorem — after Thales of Miletus, an ancient Greek thinker from about 600 BC. Legend says he was so delighted with it that he sacrificed an ox in celebration.

What made Thales special is that he did not just notice the angle was a right angle and take it on trust. He is remembered as one of the very first people to prove a geometric fact — to argue, step by careful step, that it must be true for every circle, everywhere, forever, not just the ones he happened to draw. That habit of demanding a proof is the seed the whole of mathematics grew from. Two and a half thousand years later, builders still use circle-and-diameter methods to check that a doorway or a foundation has truly square corners.