Dividing a Circle into Equal Parts

≈ 32 min

Dividing a Circle into Equal Parts

Dividing a circle underlies bolt-hole patterns, gear blanks, clock faces and every regular polygon.

Six parts is free. The radius steps around the circumference exactly six times, because each step forms an equilateral triangle with the centre. No calculation, no protractor.

Three parts is every second one of those six. Twelve is each of the six bisected.

Four parts comes from two perpendicular diameters; eight from bisecting each quadrant.

Other divisions — five, seven — need either their own construction or a calculated central angle (360° ÷ n).

Worked example. A flange with six equally spaced bolt holes needs no measurement at all: draw the pitch circle, step the radius round it six times, and drill at each step. The construction that gives a hexagon gives the bolt pattern.

Core checkpoint: The radius stepping exactly six times around a circle is the single most useful fact in this section.

How many times does the radius step exactly around the circumference of a circle?

Why does the radius step exactly six times, rather than some other number?

The chord equals the radius, so each step subtends 60° at the centre, and 360 ÷ 60 = 6.

Because six is the largest number of equal parts a circle can have.

Because compasses are manufactured to that proportion.

It is an approximation that happens to be close enough.

What central angle in degrees separates each hole of a five-hole equally spaced bolt pattern?

A learner tries to step the radius around a circle to produce five equal parts and finds it does not close. Why?

Radius-stepping only gives 60° divisions, which suit six parts; five parts requires 72°, so a different construction is needed.

The compasses were set incorrectly.

The circle was drawn too small.

Five equal parts is impossible on a circle.