Skip to content
Paper Play

Geometry & modular

Regular pentagon (golden ratio)

Find the golden ratio hidden in the square and fold an exact regular pentagon — no ruler, no scissors.

  • DifficultyChallenging
  • PaperSquare, 15 × 15 cm
  • Steps9
  • About12 min

Why this level: Many creases to make and two folds that bring a point onto a line while the crease goes through another point. Go slowly and precisely.

You need: 1 sheet of origami paper

Preview Preview: checked on screen; real-paper verification is in progress.

Start folding

Story

A regular pentagon is harder to fold than a triangle or a hexagon, because its 108° angles are tied to the golden ratio, about 1.618. T. Sundara Row’s 1893 book gives the pentagon a whole chapter and folds exactly this one: its base is 0.618 of the square’s side and it is as wide as the whole square. The way we find the golden points here (by halving an angle) is a little different from Row’s, but it leads to the same pentagon.

Did you know? You can also make a regular pentagon from a long strip of paper: tie it in a simple knot and gently pull the ends until the knot lies flat.

Learn more (external sources)

Steps at a glance

  1. 1. Two centre lines

    White side up. Fold in half side to side and unfold, then bottom to top and unfold: two centre lines.

  2. 2. Two slanted creases

    Crease a line from the middle of the left edge to the top-right corner, and one from the middle of the right edge to the top-left corner.

  3. 3. Find the golden points

    Fold the lower half of the left edge up onto the slanted crease that starts at the middle of the left edge (the new crease starts there too), and unfold. Do the same on the right. Where the new creases meet the bottom edge are the two golden points: the pentagon’s bottom corners.

  4. 4. Two vertical creases

    Fold the right part over so that the right golden point touches the vertical centre line, then unfold. Repeat on the left.

  5. 5. Fold the two lower sides

    Fold the bottom-right corner up, creasing from the right golden point, so that the corner lands on the vertical crease from step 4. Fold the bottom-left corner the same way.

  6. 6. Find the top point

    Fold the lower part up with the crease through the new corner on the right edge, so that the right golden point (the pentagon’s bottom-right corner) touches the vertical centre line. Pinch a mark there and unfold.

  7. 7. Fold the top left

    Fold the top-left part down toward you along the line from the corner on the left edge to the top mark.

  8. 8. Fold the top right

    Fold the top-right part down through all layers along the line from the top mark to the corner on the right edge.

  9. 9. Turn over: done

    Turn it over: a regular pentagon with five equal sides and five equal angles.

Settings

Clear saved data