by Arsalan Wares
Edited by Jane Rosemarin

I find the process of folding origami boxes enchanting. Origami boxes can be used to give meaningful gifts to people you care about, and you can relax while you fold these boxes. But there is more to it than that; you can derive a mathematical formula for the volume of the box from scratch! Origami, for me, always creates a context for mathematical thinking. And that’s one of the best things about origami. In this article we will learn to fold a triangular box made from six rectangular sheets. We will also discuss the exact dimensions of the box.

## The Construction Process

A picture of the box we will be making is shown above. This a simple box that is stable, sturdy and reusable. It is perfect as a small gift box. This box also looks good when folded from plain sheets. Six sheets of three contrasting colors look great.

We will need six 11-by-8½-inch rectangular sheets. Three of the sheets make each piece of the box: bottom and top. The bottom and the top pieces are identically constructed. Due to the nature of the box, the top piece expands slightly as the bottom piece is inserted. You will find a video that shows how the box is made at the end of this article, or use this link.

The video is about nine minutes long, but it will probably take about 40 minutes to make a decent box. No prior experience in origami is needed to make the box, but a little bit of mindfulness and patience can only help make the box sharper. The box shown above was made with 24-pound printer paper. We can also use 65-pound cardstock to make a sturdier version. However, I would encourage readers to make their first box with printer paper, which is easier to fold than cardstock, especially for beginners.

## The Dimensions of the Box

We used 11-by-8½-inch sheets (U.S. letter) to make the box. With the help of high-school geometry, we can establish that the base of the box is an equilateral triangle. Can you prove why this is the case? We can ask a more specific question: Can you explain why the acute angle between the first and the fourth crease is 60 degrees? Since the top part of the box expands slightly, the dimensions we discuss below pertain to the bottom part of the box. By analyzing the crease marks on the sheet, we can show that the length of each edge of the triangular base of the box is $$\frac{11}{\sqrt{3}}\approx$$ 6.35 inches and the height of the constructed box is 2¾ inches. Can you explain why? The picture below shows the box with 10 ounces of truffles. It gives you an idea of the size of the box.