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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?

See all of George Hart’s Math Monday columns

Gareth Branwyn

Gareth Branwyn

Gareth Branwyn is a freelancer writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor for Boing Boing and WINK Books. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.

3 Responses to Math Monday: Cardboard Catenary Arches

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  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:

    Great work, thanks your hardworking!

  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.

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