How do I find the centre, or divide something into equal parts, without any arithmetic?

Dividing 143mm into five is arithmetic; dividing a board into five equal parts is geometry, and the geometry is quicker, exact and impossible to get wrong. These constructions are old, they need nothing but a rule and a pencil, and they are what joiners actually do rather than reaching for a phone.

What you'll need

• A rule or a tape
• A square
• Compasses or dividers (optional)

Step by step

1. Divide a width with an angled rule.Pick a number bigger than the width that divides neatly by the number of parts you want — for five parts across roughly 143mm, use 150. Put the rule's zero on one edge and swing it until 150 sits exactly on the other edge. Mark at 30, 60, 90 and 120.
2. Square those marks off the edge.Use a square from the same edge the rule started on. Because the rule crosses at a constant angle, equal steps along it project to equal steps across the board, whatever the actual width is. It is exact, not approximate.
3. Find a centre line by marking from both ends.Measure roughly half from one end and mark; measure the same figure from the other end and mark. The true centre is exactly halfway between the two marks, and you can split that tiny gap by eye far more accurately than you can measure half of an awkward number.
4. Find the centre of a board edge with a gauge or square.Set a marking gauge or a combination square to a little under half the thickness and scribe from both faces. Two lines straddling the centre; halve the gap, reset, and repeat. Two or three passes lands on the exact centre without dividing anything.
5. Find the centre of a circle with two chords.Draw any straight line across the circle, bisect it with compasses and draw the perpendicular. Repeat with a second chord at a different angle. The two perpendiculars cross at the centre.
6. Or find it with a square.Lay a framing or combination square so the inside corner touches the rim. The two points where its arms cut the rim are the ends of a diameter — draw that line. Move the square and repeat; the two diameters cross at the centre.
7. Use dividers for repeated equal spacing.Set dividers to your best guess of the spacing and step them along. If the last step overshoots by 3mm across six steps, close the dividers by half a millimetre and go again. Two or three trials land it exactly, with no measuring at all.
8. Check the result before you commit.Measure the outer two divisions against each other, or fold a strip of paper. A construction that has gone wrong is usually wrong at one end, and one check catches it.

Tips

• The angled rule trick is the standard way to space shelf pins, balusters, fence pales, drawer pulls and screw holes. It works on any width and never produces a recurring decimal.
• For dividing a circle into equal parts, the radius steps round the circumference exactly six times. That gives you sixths, thirds and halves for free, and bisecting those gives twelfths.
• Dividers are the fastest tool here and the least owned. A cheap pair pays for itself on the first job with repeated spacing.
• When dividing a run of boards or tiles, do the arithmetic anyway to check the construction — but do the marking geometrically, because that is where errors enter.

Common mistakes

• Squaring the marks off the wrong edge — The divisions must be projected perpendicular to the edge the rule started from. Squaring from an adjacent edge distorts the spacing.
• Using a number that does not land exactly on the far edge — The construction only works if the rule's chosen figure sits precisely on the second edge. A millimetre out there is a millimetre spread across all the divisions.
• Halving a measurement in your head and marking once — It gives you one number with one error. Marking from both ends and splitting the difference cancels the error entirely.

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