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Lesson 16 of 20 · 10 moves

The Pentagon

Inscribe a regular pentagon with a corner at T.

OTR
The finished construction: the figure in gold over the lines that found it.

Ptolemy’s construction, from the Almagest. The golden cut hides inside: the side of the pentagon is the hypotenuse of a triangle with legs 1 and 1/φ.

Find the midpoint M of OR, swing MT down to the diameter at N, and TN is the pentagon’s side.

Try it in Construct

The app snaps to every crossing, checks your construction exactly, and offers hints one move at a time.

Step by step

Bisect draws the perpendicular bisector of two points in one move: it stands for the circles and line of the perpendicular bisector.

  1. OTR

    Bisect O and R: the midpoint M.

  2. OTR

    Circle centred on M through T: it cuts the diameter at N.

  3. OTR

    Circle centred on T through N: its radius is the side. It marks two corners.

  4. OTR

    Circle centred on the left corner through T: the next corner.

  5. OTR

    Circle centred on the right corner through T: the last corner.

  6. OTR

    Join the corners: all 5 sides.

What it shows

Five is the number of life: flowers, starfish, the human hand. Every diagonal is φ times a side.