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Constructions and Tilings

60°, 30° and 15° with a compass

A set square has a 60° corner and a 30° corner. A compass can draw both.

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A ray OP, and an arc about O through P.
Step 1 of 4
  1. A ray OP, and an arc about O through P.
  2. The same span from P cuts the arc at C.
  3. Rule OC. Triangle OPC has three equal sides.
  4. Bisect the corner at O. It splits into 30° and 30°.

The rule

Two arcs of one span make an equilateral triangle. Its corner is 60°. Bisect a corner to halve it. Bisect 60° to get 30°. Bisect 30° to get 15°.

Why 60°? In triangle OPC, OP and OC are both radii of that arc. A radius joins the centre to the arc. PC has the same span. So all three sides are equal: OP = OC = PC. A triangle like that is equilateral. Its three angles are equal, and they add to 180°. Each is 180 ÷ 3 = 60°.

To add 60°, strike the same span. Start where the last ray cuts the arc. The new ray is 60° further round.

Example: bisect twice to get 15°. Add 60° to that for 75°. Bisect 60° once, then add 60°, for 90°.

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This track

  1. 0 Start here
  2. 1 Copy a corner, no protractor
  3. 2 The line fair to both ends
  4. 3 Split a corner in two
  5. 4 A parallel through a point
  6. 5 Three lengths, one triangle
  7. 6 Two sides, the angle between
  8. 7 Covering a floor with no gaps
  9. 8 The corners must add to 360°
  10. 9 A square line from a point
  11. 10 60°, 30° and 15° with a compass
  12. 11 A hexagon from a circle
  13. 12 The tangram
  14. 13 Tiling the whole floor
  15. 14 Checkpoint: perpendiculars, 60°, hexagons and floors
  16. 15 The big picture
  17. 16 Track quiz

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