3D Shapes: From Cubes to Polyhedra
What Are 3D Shapes?
We live in a three-dimensional world, full of shapes we can touch, build, and move around. A ball is a sphere. A house looks like a brick. An unsharpened pencil is a cylinder, and an ice cream cone, just as its name suggests, is a cone.
As kids explore their surroundings, they begin to notice these solid forms everywhere. A cup, a box, a tower of blocks — each has height, width, and depth. Learning to recognize and name these shapes helps kids make sense of the space around them.
Why Does It Matter for Kids?
Many parents notice from experience that spatial understanding is closely linked to math success. A child who struggles with subtraction may benefit from walking backward, and one who finds number comparisons difficult might learn through physical play — crawling through tunnels, nesting pots inside each other, or jumping from objects of different heights. There’s deep neuropsychological reasoning behind this connection.
Classic research by the neuropsychologist Alexander Luria showed that the parietal regions of the brain are responsible for spatial organization, counting, and writing. When these areas are affected, both spatial and numerical skills suffer. Modern neuroscience confirms that number processing and spatial attention rely on the same brain circuits.
Studies on dyscalculia, a learning disorder that affects number sense, highlight the importance of early detection and targeted support. Spatial training is one of the key tools for helping kids overcome early learning challenges.
From a teaching perspective, spatial reasoning is the foundation for arithmetic, geometry, and even algebraic thinking. Longitudinal studies show that kids who often play with blocks, construction sets, and puzzles perform better in spatial and math tasks throughout elementary and middle school. The benefits are even stronger when adults accompany play with rich language: “We built a tall tower”, “Look, this piece goes on top,” or “This cube has a longer edge.” Talking about what kids see and build helps them connect movement, logic, and words.
How Do We Teach?
We start with the simplest shapes — those made of cubes. It’s fun to count how many cubes are in a structure, especially when some are hidden and must be imagined. Kids explore whether all cubes lie on the same level or if the structure rises across several layers.
Then we move on to other solids, such as polyhedra. Kids can build them from cardboard, or from sticks and small spheres made of clay or marshmallows. Some sets even use magnetic rods that connect to metal balls, showing how vertices, edges, and faces fit together.
Kids count these parts and compare them: faces can be triangles, squares, or pentagons. If all the faces are the same, they can’t be hexagons or octagons — those shapes don’t meet neatly at a single point. But there are solids that mix triangles and octagons, and discovering them feels like finding a hidden pattern in space.
Connecting flat and solid forms is another exciting step. A drawing might show a shadow, a side view, a top view, or an isometric picture. In the 3D Shapes & Projections game, kids rotate a solid object until its view matches a given image.
The same idea appears in the Mathcraft mission The Minifigures Experiment, where kids explore visual paradoxes and optical illusions while learning how a 3D figure can look completely different when projected onto a flat surface.
First Steps
3D Shapes Built from Cubes
We start with the simplest shapes, those made of cubes. Kids look at a drawing of a cube construction and imagine what it would look like in real space. Sometimes they just need to count how many cubes it has.
Counting with a twist
Shapes made with 5 cubes
The counting can be indirect — not all cubes are visible, so kids have to use logic, comparing the lengths of blocks and sides that stand next to each other.
When there are many cubes and some are hidden, kids begin to rely on the idea that each layer is built in the same way. This kind of layered thinking leads naturally toward multiplication — understanding that equal parts can repeat.
In some tasks, kids look at several constructions and decide which are the same and which are different. One may have more cubes. Another may look the same but always takes up two “floors,” no matter how you turn it. Sometimes they imagine what would remain if part of the structure were removed, learning to picture and transform 3D forms in their mind.
Counting with a twist
Shapes made with cubes
Counting with a twist
Shapes made with cubes
Counting with a twist
Shapes made with cubes
Counting with a twist
Shapes made with cubes
Deep Understanding
Vertices, Edges, and Faces in 3D Shapes
The most important solids in geometry are polyhedra — shapes with flat polygonal faces, straight edges, and pointed vertices. Two faces meet along an edge, and each edge connects two vertices.
In regular polyhedra, all faces have the same shape. For example, in this task kids choose the polyhedra with triangular faces — the tetrahedron, the octahedron, and the icosahedron.
2D and 3D shapes
3D shapes with triangle faces
One way to explore these solids is to glue them together from flat polygons. To make gluing easier, small flaps are added along the edges. Another way is to build them from sticks and soft materials like chickpeas or marshmallows. One kind of task asks kids to match the two models — the paper “surface” version and the stick “frame” version — and see how they fit together.
To explore different polyhedra, we invite kids to count their edges (sticks) and vertices (marshmallows). Counting vertices on a real 3D model is much harder than on a drawing. In a picture, all the corners lie on one flat surface, so you can simply move your eyes from left to right and top to bottom. But on a real model, the vertices sit on different sides. As kids turn the shape to look more closely, some corners disappear from view, and it becomes easy to lose track of which ones they have already counted. It’s also interesting to figure out which polyhedra can be built from a given set of pieces and which ones can’t.
3D shapes
Build models of 3D shapes
3D shapes
Build models of 3D shapes
Graphs
3D graphs
3D shapes
Build models of 3D shapes
Confident Mastery
Look from Above and from the Sides
The simplest way to connect a 3D object with its projection is to imagine its shadow, the shape it would cast in the light.
In this kind of task, kids learn to think of a solid as made up of smaller parts. For example, you can ask where the shadow of the red block is and where the shadow of the green one is.
Projections
Match an object to its shadow
In the 3D Shapes & Projections game, a solid object appears right inside the room through augmented reality. To see it from a certain angle, kids move around it with a tablet, looking from above, below, left, or right.
Sometimes, even a small change of viewpoint is easier to reason through logically than to imagine in your mind. In one task, kids look at an inflatable swimming ring and choose how it would appear from above. They might ask themselves: Which parts of the ring are closer to the inner circle? Which are on the outer side? How many are there, and what colors do they have?
In some tasks, kids match a room plan with its spatial view, or, as it’s often called, an isometric drawing. Others show a garden map with a marked treasure spot and a picture of the same garden. Kids may need to rotate the map in their minds to figure out which tree in the picture matches the one on the map.
Sometimes they face a classic construction challenge: imagining a part or a building by looking only at its two projections, from the top and from the side.
One of our Mathcraft missions is devoted to exploring how the same 3D structure can look completely different from different points of view.
3D shapes & projections
Rotate 3D object (House with garage) to match the given projection on a standard coordinate plane
Projections
View from above
Projections
View from above
Projections
View from above
Optical illusions
Front view and side view
Optical illusions
The minifigures experiment
Big Ideas
The Euler formula shows a beautiful relationship between the parts of any polyhedron:
V – E + F = 2,
where V is the number of vertices, E the number of edges, and F the number of faces.
This formula later becomes a doorway into topology, the study of how surfaces connect and transform.
Knowledge of polyhedra also plays a role in engineering, in the stability of structures, in crystallography, and even in molecular biology — many viruses have geometric shells.
Shadows and projections play an essential role in engineering graphics, architecture, and computer game design. Maps and plans help us navigate the world, while assembling 3D objects from flat nets connects directly to packaging, design, and even fashion.
