Geometry Problem Solver 3D geometric shapes
A 3D shape, or solid, has length, width and depth, so it takes up space instead of lying flat. The six taught first are the cube, rectangular prism, sphere, cylinder, cone and pyramid. Solids with only flat faces are polyhedra; the ones with a curved surface roll.
Faces, edges and vertices
A face is a flat surface, an edge is where two faces meet, and a vertex is where edges meet. Counting them is how solids get told apart on paper.
For any polyhedron the three counts are tied together by Euler's formula: faces + vertices - edges = 2. A cube gives 6 + 8 - 12 = 2. A square pyramid gives 5 + 5 - 8 = 2. If your count fails the check, you miscounted.
Prism or pyramid
A prism has two identical parallel bases joined by rectangles, so its cross-section stays the same all the way along. Slice a rectangular prism anywhere parallel to its base and you get the same rectangle.
A pyramid has one base and triangular faces that meet at a single point. Its cross-section shrinks as you climb. Both are named after their base: a triangular prism, a hexagonal pyramid.
Which ones roll
The sphere, cylinder and cone have curved surfaces, so they roll. The cube, prisms and pyramids have only flat faces, so they stack instead.
The cylinder is the useful edge case: lying down it rolls, standing up it stacks. A tin of food demonstrates both in about two seconds, which is why it beats any diagram for a first explanation.
Faces, edges and vertices of the common solids
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular prism | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Cylinder | 2 flat + 1 curved | 2 curved | 0 |
| Cone | 1 flat + 1 curved | 1 curved | 1 |
| Sphere | 1 curved | 0 | 0 |
Questions
What is the difference between a 2D and a 3D shape?
A 2D shape has length and width only and can be drawn complete on paper. A 3D shape adds depth, so it occupies space and can be picked up. A circle is 2D; a sphere is its 3D counterpart.
Is a cube a rectangular prism?
Yes. A cube meets every condition of a rectangular prism and adds one more: all its faces are squares. Every cube is a rectangular prism, but most rectangular prisms are not cubes.
How many faces does a sphere have?
One, and it is curved. A sphere has no flat faces, no edges and no vertices, which makes it the simplest solid to describe and the hardest to draw.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
