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bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your study of biology at A-level and university.

Mixed up Mixture

Can you fill in the mixed up numbers in this dilution calculation?

Dilution Series Calculator

Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

Packing 3D Shapes

Age 14 to 16
Challenge Level


For the following solids, try to visualise how best to pack lots of them together so as to use up the least space. Can you draw a clear diagram or give a clear explanation of your packing mechanism in each case?

  1. Spheres
  2. Circular-based cones of a fixed base radius and height. Do the relative values for these two measurements affect your packing strategy?
  3. Long chains of spheres connected flexibly by rigid rods 0-0-0-0-0- (like a long string of beads). How do the sizes of the rods/spheres affect the results
Can you think of biological situations in which the shapes of real objects are closely approximated by some of the shapes described?

Can you think of biological situations in which shapes pack together in the ways described here?

What about situations in which shapes do not pack together in these ways?

Notes and background

Whilst it might seems relatively simple, the problem of 'shape packing' is very difficult mathematically to solve with certainty. Intuitive visualisation often works just as well as a strict mathematical analysis, and often is the only sensible possibility with packing together complicated shapes.

In reality, complex molecules such as proteins pack, or fold, together in very intricate ways..