# Math Monday: Cardboard Catenary Arches

By George Hart for the Museum of Mathematics

The shape of a hanging chain is called a catenary curve.  Each link settles in to an equilibrium angle relative to its neighbors, so if you turn everything over, those angles give a plan for blocks which can rest on each other to make an arch in equilibrium. This suggests a fun construction: making giant arches from separate cardboard blocks.

Edward Ebert gives instructions here on how to hang a chain, measure it, and scale up the measurements to large cardboard blocks. If properly made, the blocks rest stably on each other without tape, glue, or clips to join them.

How high a catenary arch can you make?

More:
See all of George Hart’s Math Monday columns

### BY Gareth Branwyn

Gareth Branwyn is a freelancer writer and the former Editorial Director of Maker Media. He is the author or editor of a dozen books on technology, DIY, and geek culture, including the first book about the web (Mosaic Quick Tour) and the Absolute Beginner’s Guide to Building Robots. He is currently working on a best-of collection of his writing, called Borg Like Me.

### 3 Responses to Math Monday: Cardboard Catenary Arches

1. Anonymous on said:

“math monday”™ about Catenary curves and no equation? (i’m shocked…shocked)

how about:  y = k*cosh(x/k) = (k*(exp(x/k) + exp(-x/k)))/2

i leave it for others to post the Taylor series …yep.

2. Anonymous on said:

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3. I would love to have the plans for the catenary arch model. I am a teacher and would like to pass that learning onto my students. Thank you.