Once you have all the materials you need, decide how large you want your pentagons to be. You will need to cut out 6 pentagons and 10 smaller equilateral triangles. Using the amount of cardboard you have, estimate 6 square blocks of the same size that will be used to cut the pentagons. You'll also need some extra cardboard for for the ten smaller triangles, an additional 2-3 square blocks is enough for your triangles, depending upon how you cut out your triangles. Try to plan your shapes on the cardboard such that any folds in the cardboard are avoided. The folds will weaken the structure. Once you have gathered all your materials, begin calculating the size of your shapes.
Measure the length of the square blocks that you have reserved for creating the pentagons. The length of each side of the pentagon should be .58333 times the length of your block. For example, if the block is 3'x3', or 36"x36", the length of each pentagon side will be 1.75' or 21". Write down the length of the sides of your pentagon. A pentagon is a 5 sided polygon where all sides are the same length and all five internal angles are 108 degrees.
Draw the first side of the pentagon centered at the bottom of the block. Use a protractor to measure 108 degrees from the left edge of the first line and draw a line the same length. Repeat this for drawing all sides of the pentagon.
An adult can then cut out the pentagon by using a box cutter (recommended) with extra scrap cardboard padding underneath so the lower surface doesn't get scratched, or use a strong pair of scissors. Then trace the pentagon onto the other five blocks and cut the pieces out.
Use the same length you used for the pentagon sides to make the ten equilateral triangles. An equilateral triangle has three equal sides and three internal angles of 60 degrees. An adult can cut out the triangles using a box cutter (recommended) with extra scrap cardboard padding underneath. The first cut-out triangle can be used to trace the remaining triangles.
Once you have all 6 pentagons and 10 triangles cut out, assemble the pieces into a dome shape using duct tape. Follow the pattern in the picture to assemble the geodesic dome. Start by assembling pentagons and triangles around in a circle, leaving the top pentagon for the last piece. One triangle can be used for a door and should only be attached on one side so it can open and close. Alternatively, a pentagon can be used for a door and should only be attached on one side so it can open and close. A pentagon door is a larger door, but the structure will not be as strong. The duct tape allows for some flexibility and room for error if the pieces are not cut to exact specifications. One or both sides of cardboard can be duct taped together.
This is an upside-down view of the geo-dome. It is included here to show you how to assemble the pentagons and triangles.