I started work on a custom 6-sided steel glassblowing mold, based on a pyramid (faceted cone).

In the process I uncovered a** quick and easy math-based craft project **which you may find suitable for your elementary school (or older) children. My brother’s kids seemed to enjoy this project yesterday during Thanksgiving.

Basically, with only a protractor, scissors or paper cutter, ruler, and a few pieces of tape, you and your kids can create n-sided pyramids. In other words, 3, 4, 5, or 6-sided pyramids that are fun to make, and if decorated, can make some really nice paper hats, kind of like Wizard Hats! As well as perhaps teaching a little bit of math along the way!

This project started from the wonderful online article Compound Angles – Calculations and Jig for Making for N-sided Pyramids by Tony Beugelsdijk of the Albuquerque Woodworkers Association. I have an extra copy of that PDF stored

here (if you have a problem with the first link).

There is some really interesting math — algebra and trigonometry — which I’ll go over at the end of the article, if you are interested, but for now, here is a summary table if your goal is to have some math-based fun with a kids craft project, and make some hats!

Number of Sides | Width (inches) of Base Side | Height of Pyramid (hat) | Angle to be Measured |

4 sides | 6 inches | 11 inches (letter-size paper) | 15 degrees |

6 sides | 5.5 inches | 11 inches (letter-size paper) | 14 degrees |

6 sides | 4 inches | 11 inches (letter-size paper) | 10 degrees |

5 sides | 5 inches | 11 inches (letter-size paper) | 13 degrees |

5 sides | 4 inches | 11 inches (letter-size paper) | 10 degrees |

I’m going to make the last one — a 5-sided hat which is 4 inches on each side and 11 inches tall.

- Start with the protractor on the corner of a piece of standard letter-size paper,
**and place a dot at 10 degrees.**

- Draw a 5-inch line along the dot to the corner of the paper

- Draw another line from the point where you ended to the other corner of the paper.

- You will need to hold a stack of 5 pieces of paper, and cut out the isosceles triangles you just drew. This step is especially easy with a paper cutter

- You will then hopefully have five identical triangles

- Which you need to tape together their long sides.

- Now (while flat) would be a good time to decorate the sides!

- Tape the final edges together, you’ve got a beautiful hat!

If you would prefer a flat-topped hat, you could create a truncated pyramid (Pyramidal Frustum), using sides which look more like this (for a 6-sided two-inch-to-four-inch truncated pyramid).

How are the angles in the table above calculated? Glad you asked! Tony Beugelsdijk’s article gives worked-through examples of the math, and from those, I created this Excel spreadsheet (also available in Excel 97-2003 format) which looks like this (from the last example in the table, and the hat we made above):

In Excel 2007, typing the keyboard shortcut (toggle) ctrl+` (that is the “Grave Accent” key which is above the tab key), shows the formulas:

A1: n

B1: 5

C1: number of sides

A2: h

B2: 13.6

C2: vertical height of pyramid (not length of side)

A3: a

B3: 5

C3: base length

A4: Theta θ

B4: =ATAN(SQRT((2*h/a)^2+k^2))*(180/PI())

C4: or

D4: =90-theta

E4: degrees

A5: Phi Φ

B5: =ATAN((2*h)/(a*k))*(180/PI())

C5: first angle to be cut on the table saw

A6: b

B6: =SQRT(h^2+(a*k/2)^2)

C6: lenth of side (face altitude), this is a piece of legal-size paper

A7: k

B7: =TAN(PI()/2-PI()/n)

C7: used in various calculations