Unit 9: Circles

The parts of a circle

A circle is every point the same distance from the middle.

That distance is the radius. All the way across through the middle is the diameter, which is exactly twice the radius.

All the way round the outside is the circumference.

The number that never changes

Take any circle. Measure round it, then measure across it, and divide.

You always get the same number: a bit more than 3. We call it pi.

Nobody invented that. It is a fact about round things, and it is why one formula works for every circle there has ever been.

The circumference formula

Since circumference ÷ diameter is pi, the circumference must be pi × diameter.

And the diameter is twice the radius, so:

circumference = 2 pi r

Leave pi in your answer rather than turning it into 3.14 — an exact answer is better than a rounded one.

Area of a circle

Cut a circle into thin slices and lay them alternately up and down. They fit together into something very close to a rectangle.

Its height is the radius r, and its length is half the way round, pi r.

So the area is pi r². Not a formula to memorise — a rectangle in disguise.

Double the radius, four times the area

Area uses r^2, so doubling the radius does not double the area — it multiplies it by 4.

A 12-inch pizza has four times the pizza of a 6-inch one, not twice. This is worth knowing before you next order lunch.

From the circumference to the radius

You know C = 2 pi r. If you are told the circumference instead, that is just an equation from unit 3.

A circle has circumference 12pi. Then 2 pi r = 12 pi, so r = 6.

Divide by 2 pi and you are done. The pi cancels, which is exactly why leaving it in your answers pays off.

From the area to the radius

Area needs a square root on the way back. A = pi r^2, so r = sqrt(A/pi).

A circle of area 64pi has r^2 = 64, so r = 8.

Only the positive root counts. A radius of -8 is not a circle.

Chord, tangent, arc, segment

A chord joins two points on the circle. The longest chord of all is the diameter.

A tangent touches at one point only and never goes inside.

An arc is a piece of the edge. A sector is a pizza slice; a segment is what a chord cuts off, like the crust end.

A tangent meets the radius at a right angle

Draw the radius out to the point where a tangent touches. Those two lines are always perpendicular.

That single fact turns most tangent problems into Pythagoras problems, because you suddenly have a right-angled triangle to work with.

It also means the shortest distance from the centre to a tangent is exactly the radius.

A radius through the middle of a chord

Drop a line from the centre to a chord so that it hits at right angles. It always lands exactly halfway along.

So half the chord, the distance from the centre, and the radius make a right-angled triangle.

That is how you find a chord you cannot measure directly.

One turn covers one circumference

Roll a wheel through exactly one full turn and it travels the length of its own circumference.

A wheel of radius 30 cm covers 2 × pi × 30 = 60pi cm in one revolution — about 188 cm.

So distance travelled is circumference × number of turns, and that is the whole of it.

The rope round the earth

A famous puzzle. A rope fits exactly round the earth. Add one metre to it and lift it evenly all the way round. How high does it sit?

About 16 centimetres — enough to crawl under.

The answer does not depend on the earth at all. Adding 1 to 2 pi r adds 1/(2 pi) to the radius, whatever r was. Do it round a tennis ball and you get the same 16 cm.

A circle inside a square

A circle drawn inside a square, touching all four sides, has a diameter equal to the side of the square.

So a square of side 10 holds a circle of radius 5.

Getting that link right is the whole problem. After it, both areas are one line each.

A square inside a circle

Turn it round: a square drawn inside a circle has its corners on the edge, so the square’s diagonal is the circle’s diameter.

A circle of radius 5 has diameter 10, so the square’s diagonal is 10 and its side is 10/sqrt(2).

Diagonal for inside, side for outside. Mixing those up is the classic mistake here.

Trapping the circle

Archimedes could not measure a curve. So he did not try.

He drew a polygon inside the circle and another outside it, and measured those instead. The circle is trapped between them, so pi is trapped between their two answers.

Then he doubled the number of sides. And again. And again.

A hexagon gives a first estimate

A regular hexagon inside a circle has six sides, each exactly equal to the radius — that is the equilateral-triangle fan from unit 13.

So its perimeter is 6r, while the circle’s is 2 pi r.

The circle is longer, so 2 pi r > 6r, which means pi > 3. Six sides, and you already know pi is more than 3.

Ninety-six sides

Archimedes kept doubling until he reached 96 sides, by hand, without algebra or decimals.

He proved pi lies between 223/71 and 22/7 — that is between about 3.1408 and 3.1429.

The true value is 3.14159…, comfortably inside. Over two thousand years ago, with no calculator and no zero.

A fraction of a circle

A slice of pizza is a sector. The curved crust along its edge is an arc.

Both are worked out the same way: find what fraction of the whole circle you have, then take that fraction of the circumference or the area.

A 90° slice is 90/360 = 1/4 of the circle. So it has a quarter of the area and a quarter of the way round.

A ring between two circles

To find the area of a ring — a circle with a smaller circle cut out of the middle — do not measure the ring.

Work out the big circle, work out the small one, and subtract.

This "whole take away the hole" habit will solve a great many geometry problems.