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Pop-Up Dodecahedron

Make a 3-D geometric figure using just cardboard, and a rubber band! Then watch it pop!

Ages: 5+ (with help)
Players: 1
Time: 20+ Minutes
Type: arts and crafts
Location: tabletop
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Ages: 5+ (with help)
Players: 1
Time: 20+ Minutes
Type: arts and crafts
Location: tabletop

Instructions

Start by cutting a strip of paper about an inch thick.

Drawing a line across a paper with a ruler
Cutting across the line to make a strip of paper

(Depending on your rubber band and cardboard, you may find that a slightly larger or smaller strip of paper is ideal. We used a rubber band with a diameter of about 2.5 inches.)

Rubber band of diameter 2 and a half inches

Did you know that a paper knot, flattened out, makes a pentagon? Tie your strip into a knot, and pull the ends of the paper gently until the knot gets fairly tight. Then press the knot flat. As you press it, you may need to continue fiddling with the knot to take out any slack. Fold over the extra strips of paper until all you've got left is a pentagon.

Tying a knot with a strip of paper
Flattening the paper knot

Now you'll trace the pentagon onto a piece of thick cardboard. The sturdier, the better. Make the pattern below by tracing the pentagon six times.

Tracing the paper pentagon onto cardboard
Pentagons traced onto cardboard

Now, cut out the shape. You can do this by first making 5 straight cuts.

Cutting out the pentagon shapes
Cutting out the pentagon shapes

Look! You've made a bigger pentagon! Now cut out the 5 small triangles.

Cutting out the little triangles between pentagons

Finally, use a box cutter or other sharp knife to score the remaining lines, shown below. Be careful! This part should definitely be done by an adult.

Pentagon shapes next to a knife
Knife scoring a cut between pentagons

As you cut, be sure not to cut all the way through the cardboard. Cut only through the top layer. Then fold gently along each scored line.

Folding over the scored pentagons

That was fun! Now make a second of these shapes. (Be sure to use the same paper knot so that both are the same size.)

Two cardboard halves of a dodecahedron

This next part can be a bit tricky. Line up the two pieces you've made like this:

Two halves of a dodecahedron lined up

Now, weave the rubber band around the points of these overlapping shapes, in an over-under pattern.

Rubbber band wrapped around the points of the two pentagon shapes

Now let the rubber band pop your dodecahedron into shape!

Pushing the cardboard pentagons into a dodecahedron
Cardboard dodecahedron with a rubber band around it

Note that, if your rubber band is too tight, it may cause your dodecahedron to collapse, and if it's too loose, it may not hold the dodecahedron together.

Have fun! Squish the dodecahedron together, then let it pop back up! Kids will undoubtedly go a bit crazy with this geometrical delight, so be ready to re-weave the rubberband if it comes off.

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

Variations d

Pencil Cup:

Our dodecahedrons got less poppy over time. That's okay. We made more! But what can be done with a flabby dodecahedron? Here's an option. Cut off one of the pentagons from one of the pieces.

One pentagon cut from a cardboard cutout of six pentagons

Then tape everything together (instead of using a rubber band to hold them together). Voila!

Cardboard dodecahedron with top face missing
Pencil cup made of a cardboard dodecahedron
It could use a little paint!

Dragon Egg:

Another option is to tape together the top piece and the bottom piece separately, making two cup shapes.

Two halves of a cardboard dodecahedron

Then tape together one edge of the top piece to one edge of the bottom piece, making a hinge. Your child won't have any trouble finding things to put in their new dodeca-box. Or, find a small toy and let it hatch out!

Dodecahedron with a dragon coming out
It isn't just dragons that hatch from dodecahedron-shaped eggs!

Classroom Tips d

Make some time for Pop-Up Dodecahedron when learning about regular polygons or 3-D geometry. Make a prototype first to check that the rubber bands you have available work well with the size of paper strip you begin with.

Discussion Questions

  • Do you need regular pentagons to make this?
  • How many faces/vertices/edges does the dodecahedron have? Are these numbers related? (see Learning Notes)
  • Can you find any other polyhedra in the classroom? How do their faces/vertices/edges compare to the dodecahedra?

Alignment with Beast Academy Curriculum

  • Level 3, Chapter 1: Shapes
  • Level 5, Chapter 1: 3D Solids
  • Level 4, Chapter 1: Shapes

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

Learning Notes d

Tiling the Plane:

When we trace our pentagon onto the cardboard, notice that there are little gaps between the outer pentagons.

Pentagons on cardboard

When shapes fit together snugly, with no space between, and can cover an entire piece of paper (no matter how big), this is called "tiling the plane." (A "plane" is the math term for a flat 2-dimensional surface, like a piece of paper.)

Let your child try tracing their pentagon so that the shapes are right up against each other (no overlap) and there are no gaps between them. Don't make them try too long, because they'll quickly discover this is impossible.

So what shapes can tile the plane? Cut out different shapes to try. Also see our BA Playground game Cookie Cutter for more on this topic.

Euler's Formula:

A 3-D shape like your Pop-Up Dodecahedron is called a polyhedron. Leonhard Euler, considered one of the greatest mathematicians of all time, discovered a simple formula about polyhedra that even young math beasts can easily verify.

How many pentagon-shaped flat pieces does your dodecahedron have? (It has twelve.) The math term for these flat pieces is "faces." One way to think of the number of faces is to imagine that your dodecahedron were a die. How high would you be able to number it?

Now count the number of "vertices" your dodecahedron has. These are the pointy corners that poke out. There are a lot, so it may help to color them with a marker as you go. There are 20 vertices.

Finally, count how many "edges" it has. These are the straight-line pieces where the pentagons meet. Again, there are quite a few, so keep track by marking them with a marker as you count.

Dodecahedron with edges colored blue and vertices colored orange
The vertices are marked in orange, and the edges are marked in blue.

Your dodecahedron has 30 edges.

Now, what do you get when you add up the number of faces and vertices, then subtract the number of edges? Our calculation is: 12 + 20 - 30 = 2.

To see why this is remarkable, find some other polyhedra to examine. You may not realize this, but you likely have some other polyhedra around your home. A die is a cube, which is a kind of polyhedron. Count the number of faces, vertices, and edges. A cube has 6 faces, 8 vertices, and 12 edges. Again, add the number of faces and vertices, then subtract the number of edges: 6 + 8 - 12 = 2. We got 2 again?

You may find more polyhedra. Here are a couple more we found.

Red block and orange block

The red block has 5 faces, 6 vertices, and 9 edges. Euler's Formula gives us 5 + 6 - 9 = 2. The orange block has 8 faces, 12 vertices, and 18 edges. Euler's Formula gives us 8 + 12 - 18 = 2. You'll get 2 every time!

Be careful when searching for polyhedra. Euler's Formula only applies to what are called "convex" polyhedra, meaning shapes that don't have any parts that dent inward. Any block that has a curvy shape is also not a polyhedron. Here is an example of a couple of shapes that wouldn't work.

Green block and blue block
The green block has a rounded edge. The blue block has an indent.
Euler's Formula would not apply to either of these blocks. (Check for yourself!)

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Materials
  • paper
  • scissors
  • cardboard
  • penci
  • box cutter
  • rubber band
Learning Goals
  • wonder
  • shapes
Common Core Standards
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Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book