Fold the cup one step at a time and pause whenever you like. Ask me about any fold: why the diagonal is a line of symmetry, why the folded edge must be level, or why the small triangle is a copy of the big one.

Paper Cup

Fold a traditional paper cup from one square and find the geometry in every crease

Start with a square piece of paper and fold it corner to corner. The fold is a line of symmetry: the two halves match exactly, so everything you do on one side has a twin on the other. The triangle you get has angles of 45°, 45° and 90°, and its long edge is the diagonal of the square, about 1.41 times as long as a side.

Turn the triangle so the long edge is at the bottom. Fold the left corner over to the right edge so that the folded edge is level, which means parallel to the bottom. Parallel lines never meet and stay the same distance apart, so a level edge is at the same height all along. Fold the right corner the same way and the two folded edges land on one straight line.

Look at the little triangle that sticks up above that level line. Its angles are 45°, 45° and 90° again, so it is similar to the big triangle: the same shape, but every length is about 0.41 times as long (exactly √2 − 1). Its area is only about 0.17 of the big one, because area shrinks with the square of the length.

Last, fold the top corner of the front layer down and the top corner of the back layer behind, both along the level line. The two flaps are mirror twins, one in front and one behind, and the level line becomes the rim of the cup. Pause the animation on any step and ask the tutor why the folded edge has to be parallel to the bottom, or how to spot the 45° angles.

Related Topics

Paper Soldier CapCongruence & SimilarityPythagorean Theorem
How do you make a paper cup from a square piece of paper?
Fold the square corner to corner to make a triangle, with the long edge at the bottom. Fold the left corner to the right edge and the right corner to the left edge so both folded edges are level. Then fold the top corner of the front layer down and the top corner of the back layer behind. Each fold on the left has a matching fold on the right.
Why does the folded edge have to be level?
Level means parallel to the bottom edge. When both corners are folded so their edges are level, the two folded edges lie on one straight line. That line is the rim of the cup, so the cup stands straight and the mouth is flat.
Which maths is hidden in a paper cup?
The first fold is a line of symmetry. The triangle has 45°, 45° and 90° angles. The level folds make parallel lines, and the small triangle above the rim is similar to the big one, about 0.41 times as long and about 0.17 as much area.
Does a folded paper cup hold water?
For a little while. Ordinary paper soaks up water and goes soft, so it is best for dry snacks or a short sip. Waxed or coated paper lasts longer. Sharp, straight creases help the cup keep its shape.
What size of paper is best?
Any square works. Start with a sheet about 15 cm (6 inches) on a side, which makes a cup that fits in a child's hand. A big square makes a big cup, and the angles and ratios stay exactly the same.