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Geometry & modular

The largest equilateral triangle

One corner on the centre line makes 60°. Halve the angles next to it and fold the biggest equilateral triangle a square can hold.

  • DifficultyIntermediate
  • PaperSquare, 15 × 15 cm
  • Steps7
  • About5 min

Why this level: Every crease starts exactly at one corner; keeping the folds on that corner is the trick.

You need: 1 sheet of origami paper

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Story

An equilateral triangle has three equal sides and three 60° angles. T. Sundara Row’s book Geometric Exercises in Paper Folding, published in India in 1893, folds one from the kindergarten paper square in just this way: swing the bottom edge around one corner until the other corner rests on the middle crease. Finding the biggest equilateral triangle that fits in a square is a well-known geometry puzzle; the winner shares a corner with the square and leans 15°.

Did you know? Row was not a professional mathematician: he worked as a tax collector in southern India and did mathematics as a hobby. His book became known in Europe after the famous mathematician Felix Klein mentioned it in 1895.

Learn more (external sources)

Steps at a glance

  1. 1. Crease the centre line

    White side up. Fold the left edge to the right edge and unfold to make a vertical centre line.

  2. 2. Corner onto the centre line (60°)

    Fold the bottom-right corner up onto the centre line, with the crease starting exactly at the bottom-left corner. The folded edge now makes 60° with the bottom.

  3. 3. Halve the left angle (75°)

    Fold the left edge down to lie exactly along the folded edge, again creasing from the bottom-left corner. This halves the 30° that was left.

  4. 4. Unfold the bottom corner

    Unfold only the bottom corner from step 2. Leave the top-left part folded.

  5. 5. Halve the bottom angle (15°)

    Fold the bottom edge up onto the crease from step 2, creasing from the bottom-left corner. The two new fold lines are now exactly 60° apart.

  6. 6. Fold the last corner

    Fold the top-right corner down through all layers along the line joining the two points where the folds meet the top and right edges.

  7. 7. Turn over: done

    Turn it over: the largest equilateral triangle you can fold from the square. Measure the three sides and compare!

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