Sacred geometry · Fold it yourself
Platonic Solids 3D
Every solid can be cut along some of its edges and laid flat as one piece of paper. That flat shape is its net. Pick any of the five Platonic solids, spin it, unfold it into a net and fold it back up, then print the net and build the real thing. Turn on the dual to see the solid hiding inside it.
Cube
- Faces
- Edges
- Corners
- At each corner
- Edges you cut
- Dual
How it works
Every solid is a folded net
A net is a flat shape that folds up into a solid with no gaps and no overlaps, like a cereal box before it is glued. Slide the fold bar and watch the faces swing open on their hinges until the whole solid lies flat, then fold it back up.
Every solid has more than one net. The cube has 11, the octahedron 11, the tetrahedron only 2, and the dodecahedron and icosahedron 43,380 each. Press New net to pick hinges at random and get a different one. Albrecht Dürer printed some of the first nets like these in his 1525 book on measuring with compass and ruler.
You always cut one edge fewer than the solid has corners. A net of F faces needs F − 1 hinges to hold together, so the rest of the edges get cut: E − (F − 1). Euler’s rule V − E + F = 2 turns that into V − 1. A cube has 8 corners, so you always cut 7 of its 12 edges, whichever net you pick. The Edges you cut line shows it for every solid.
The dual sits inside. Put a corner at the middle of every face and join them, and you get another Platonic solid: the cube and octahedron swap, the dodecahedron and icosahedron swap, and the tetrahedron turns into itself. Close the solid and press Show dual.
Each solid has its own page with a calculator for volume, surface area and angles: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. Why there are only five is on the Platonic solids page.
For classrooms
Print a net and build it
Press Print this net and the net prints at real size, scaled to fit a sheet of Letter or A4 paper, with the edge length printed underneath. Solid lines are cuts, dashed lines are folds, and grey tabs carry the glue. Card stock folds best. Pick a random net first and every student can build the same solid from a different net.
Questions
Nets of the Platonic solids
What is the net of a solid?
A flat shape made of the solid’s faces, joined edge to edge, that folds up into the solid with no gaps or overlaps. A cube’s best-known net is a cross of 6 squares.
How many nets does each Platonic solid have?
The tetrahedron has 2, the cube 11, the octahedron 11, and the dodecahedron and icosahedron 43,380 each. Duals always have the same number.
Can I print the nets of the Platonic solids?
Yes. Pick a solid, press Print this net, and it prints at real size on Letter or A4 with cut lines, dashed fold lines and grey glue tabs. Pick New net first to print a different layout.
Why do you always cut one fewer edge than there are corners?
A net of F faces needs F − 1 hinges to stay in one piece, so E − (F − 1) edges are cut. Euler’s formula V − E + F = 2 makes that V − 1. A cube, with 8 corners, always has 7 edges cut.
What is the dual of a Platonic solid?
The solid you get by putting a corner at the middle of every face. The cube and octahedron are duals, the dodecahedron and icosahedron are duals, and the tetrahedron is its own dual.
Can I use this tool in my class or on my site?
Yes, free. Print as many nets as you need, and teachers and bloggers can embed the whole tool with the code at the bottom of this page.
Sources
Where this comes from
- Eric W. Weisstein, “Net,” MathWorld, for the number of nets and Dürer’s 1525 treatise: mathworld.wolfram.com.
- OEIS A201187, “Number of distinct planar nets of the Platonic solids”: oeis.org.
- Eric W. Weisstein, “Platonic Solid” and “Polyhedral Formula,” MathWorld: mathworld.wolfram.com.
- Eric W. Weisstein, “Cube,” MathWorld, with all 11 cube nets: mathworld.wolfram.com.
- J. J. O’Connor and E. F. Robertson, “Albrecht Dürer,” MacTutor: st-andrews.ac.uk.
- Euclid, Elements XIII.18, translated by T. L. Heath, Perseus Digital Library: tufts.edu.
Share
An image of your solid. Copy link reopens the same net and fold.
Poster
Your solid, printed large.
Put this tool on your site
Embed Platonic Solids 3D
Teachers and bloggers can drop the tool into any page. Paste this code where you want it to appear.