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How to Make a Geometric Net of a Rhombicosidodecahedron

A geometric net is a flat figure that can be folded into a 3-D shape. If you draw four triangles around a square in the correct way, for example, then you can cut out the shape and fold it into a pyramid. To make geometric nets of relatively simple shapes -- cubes, pyramids, cylinders -- you can usually just think about the shape for a minute and then sketch the net without too much trouble. Trying to do this with a rhombicosidodecahedron, however, is more difficult. Because a rhombicosidodecahedron is made of 20 triangles, 30 squares and 12 pentagons, it's almost impossible to think of the relationships between all the faces at the same time. To be able to draw the net, approach the problem systematically.

Things You'll Need

Ruler
Protractor
Scissors

Instructions

Preparing to Draw the Net
- 1
  Choose the side length. Choose only once because each side of each face of the rhombicosidodecahedron has the same length. Because the net will be large, ensure you have paper big enough to hold it. If you choose a side length of 1 inch, for instance, then the net will be about 8 1/2 by 14 1/2 inches, which is too large to fit on an ordinary sheet of notebook paper.
- 2
  Draw an equilateral triangle, a square and a regular pentagon on paper with the side lengths you chose. Use a ruler and a protractor to make the shapes, which will be the net's building blocks of your net. Cut each shape from the paper.
- 3
  Break the problem into small pieces. If you ignore the top and bottom pentagons on the rhombicosidodecahedron, you can divide the shape up into five identical strips, each made of six squares, four triangles and two pentagons.
Drawing One Strip of the Rhombicosidodecahedron
- 4
  Draw a triangle with one of its corners pointing straight up. Draw by tracing the triangle you drew and cut out previously, and use the other shapes you drew and cut out as templates each time you draw a shape. Draw a square below the triangle. In geometric nets, all of the shapes touch each other. From this point, draw all shapes so that they share an edge with an existing shape. When you draw a square below a triangle, the square's top edge is the triangle's bottom edge. Draw a pentagon to the right of the square. Draw another triangle below the square. Draw a square below and to the right of that triangle.
- 5
  Draw a square above and to the right of the first triangle you drew. Draw a triangle above and to the right of the square. Below and to the right of the triangle, draw a square. Make a small mark in the corner of that square to help you remember it because it's where the next strip will attach to this strip.
- 6
  Draw a square above the last triangle you drew. It's at the top of the figure, pointing down. Draw a pentagon to the left of the square. Draw a triangle above the square. Draw a square above and to the left of the triangle.
- 7
  Verify that the completed strip has six squares, four triangles and two pentagons.
Completing the Net
- 8
  Draw a triangle below and to the right of the square in which you placed a small mark. This will be the first triangle of the second strip, which is exactly the same as the first strip. Follow the instructions from Section 2 to draw the second strip.
- 9
  Draw new strips in this way three more times, creating a total of five strips.
- 10
  Add the top and bottom pentagons to the figure. Put one of the pentagons above and to the right of one of the topmost squares, and place the other pentagon below and to the left of one of the lowest squares.
- 11
  Check your work. It should have five identical strips of shapes, each strip containing two pentagons, four triangles and six squares. It also should have two additional pentagons, one at the top and one at the bottom of the figure. The result is a geometric net of a rhombicosidodecahedron.

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