The 4 sided pyramid actually has two definitions--whether or not you count the base as a "side." By geometric definition, a 4-sided pyramid has a triangular base with three sides, and when the sides and base are all equilateral triangles, it is called a tetrahedron. The square-base pyramid, like the Pyramid of Giza, is also called a four-sided pyramid, because it has four equal vertical sides. Either four sided pyramid can be made with sticks, balls, sheets, or solid materials.
Decide how many layers you want to make into a triangle-based pyramid. Two layers consist of a single ball on the top layer, and three balls on the second layer. Three layers consist of the same first two layers, plus six balls in the third layer.
Determine the number of balls you need. Each lower layer has the same number of balls as the next upper layer, plus a number of balls equal to the layer level below the top.
To figure how many balls for each layer, consider: the top layer is one ball. The second layer is three balls, or the first layer plus two balls. The third layer is six balls, or the second layer plus three balls. The fourth layer is 10 balls, or the third layer plus four balls. Two layers need 4 balls; three layers need 10 balls; four layers need 20 balls.
Lay the bottom layer. Arrange the bottom layer in an equilateral triangle shape (all three sides equal). You can provide a border to hold the balls together, or glue the balls together.
Lay the next layers. The balls will fit between the balls in the bottom layer. Continue laying the balls to finish the tetrahedral pyramid. You can glue the upper layers or just let them sit by gravity.
Decide how many layers you want to make the pyramid. This one is easy--each level is the square of sequential numbers.
Determine the number of balls you need to make the pyramid. The number of balls for each layer, from the top, is 1 x 1; 2 x 2; 3 x 3; 4 x 4; and so on. The seventh layer would have 7 x 7 balls or 49 balls. The ninth layer would have 9 x 9 or 81 balls, and so on. Add up the number of balls in each layer.
Lay the base layer. Put a square border around it to keep the balls from rolling away, or glue the balls together.
Add the additional layers. The top layers will be held together by the base layer and don't have to be glued, but you can glue them if you want. Civil war soldiers made "cannon ball pyramids" in this shape.
Get four sticks each identical length for a triangular pyramid. For a square pyramid, use eight identical sticks.
Lay out three sticks in a triangle base for a triangle pyramid, and join them together (glue, tape, any way). Attach the other three sticks at the corners, and bring them together at the top to form the pyramid.
For a square pyramid, lay out four sticks for the square base and join them. Attach the other four sticks at the corners, and bring them together at the top to form the pyramid.
Use identical lengths for all edges for sheet or solid pyramids. A tetrahedral pyramid is made of four identical equilateral triangle shapes. Join each triangle to the next to form the pyramid.
For the square pyramid, start with a square base. Use the length of a side to determine the size of your triangular sides. Cut each triangle with the edges equal to one side of the square. Join the sides to the base, then join the sides of each triangle to form the pyramid.
Bevel solid sheets like wood so they fit together cleanly. The sides of a square pyramid fit together at 76 degrees--not 90 degrees like you might expect. Set the angle to cut the edges at 38 degrees.
The triangle pyramid edges fit together at about 70 degrees. Set the angle to cut at 35 degrees.