Shapes in Geometry — Complete 2D and 3D Taxonomy
Shapes in Geometry — Complete 2D and 3D Taxonomy
TL;DR
Geometric shapes split into 2D (flat, with length and width) and 3D (solid, with length, width, and height). 2D shapes group into polygons (triangles, quadrilaterals, regular and irregular) and non-polygons (circle, ellipse). 3D shapes group into polyhedra (prisms, pyramids, Platonic solids) and curved solids (sphere, cylinder, cone).
The Top-Level Split — 2D vs 3D
| Aspect | 2D shapes (planar) | 3D shapes (solid) |
|---|---|---|
| Dimensions | Length and width (no thickness) | Length, width, and height |
| Measurement | Area (square units), perimeter (linear units) | Volume (cubic units), surface area (square units) |
| Where they appear | Drawings, maps, blueprints, plane geometry | Real-world objects, physical models, solid geometry |
| Example | Square, circle, triangle, hexagon | Cube, sphere, cylinder, cone |
Every 3D shape's faces are 2D shapes. A cube's faces are squares; a tetrahedron's faces are triangles; a cylinder's curved face unrolls into a rectangle. So 2D shapes are the building blocks of 3D shapes.
2D Shapes — Complete Taxonomy
The two top-level 2D families
| Family | Description | Examples |
|---|---|---|
| Polygons | Closed figures with straight sides | Triangle, quadrilateral, pentagon, hexagon |
| Non-polygons (curved) | Closed figures with at least one curved boundary | Circle, ellipse, semicircle, oval |
Triangles (3-sided polygons)
| Type | By sides | By angles | Properties |
|---|---|---|---|
| Equilateral | All three sides equal | All three angles 60° | 3 lines of symmetry, 120° rotational |
| Isosceles | Two sides equal | Two angles equal | 1 line of symmetry |
| Scalene | All sides different | All angles different | No symmetry |
| Acute / Right / Obtuse | Side classification independent | Independent angle classification | Combined with side type to give multiple types |
Triangle area: (\frac{1}{2} \times \text{base} \times \text{height}) for any triangle. Heron's formula handles cases with all three sides given.
Quadrilaterals (4-sided polygons)
| Type | Defining property | Sides | Angles | Area formula |
|---|---|---|---|---|
| Square | All sides equal, all angles 90° | 4 equal | 4×90° | s² |
| Rectangle | Opposite sides equal, all angles 90° | 2 pairs equal | 4×90° | l×w |
| Rhombus | All sides equal, opposite angles equal | 4 equal | Opposite pairs equal | (\frac{1}{2} d_1 d_2) |
| Parallelogram | Opposite sides parallel and equal | 2 pairs equal | Opposite pairs equal | base×height |
| Trapezoid (US) / Trapezium (UK) | One pair of parallel sides | Variable | Variable | (\frac{1}{2}(a+b) \times h) |
| Kite | Two pairs of adjacent equal sides | Two pairs adjacent equal | One pair of equal angles | (\frac{1}{2} d_1 d_2) |
Every square is a rhombus and a rectangle. Every rectangle is a parallelogram. Every rhombus is a parallelogram. The quadrilateral family forms a hierarchy of inclusion.
Polygons with more than 4 sides
| Polygon | Sides | Interior angle (regular) |
|---|---|---|
| Pentagon | 5 | 108° |
| Hexagon | 6 | 120° |
| Heptagon | 7 | ≈128.57° |
| Octagon | 8 | 135° |
| Nonagon | 9 | 140° |
| Decagon | 10 | 144° |
A polygon is regular when all sides and all angles are equal. Each interior angle of a regular n-gon is (\frac{(n-2) \times 180°}{n}).
Non-polygons (curved 2D shapes)
| Shape | Description | Key formula |
|---|---|---|
| Circle | All points equidistant from a centre | Area = πr², Circumference = 2πr |
| Ellipse | Stretched circle with two foci | Area = πab (semi-major a, semi-minor b) |
| Semicircle | Half a circle | Area = (\frac{1}{2} \pi r^2), Perimeter = πr + 2r |
| Oval | Egg-shape (not formally defined) | — |
3D Shapes — Complete Taxonomy
The two top-level 3D families
| Family | Description | Examples |
|---|---|---|
| Polyhedra | Solids with flat polygonal faces | Cube, tetrahedron, octahedron, prism, pyramid |
| Curved solids | Solids with at least one curved face | Sphere, cylinder, cone, torus |
Prisms (two parallel polygonal bases, rectangular lateral faces)
| Prism | Base shape | Volume | Faces |
|---|---|---|---|
| Triangular prism | Triangle | (\frac{1}{2} \cdot b \cdot h_t \cdot L) | 5 (2 triangles + 3 rectangles) |
| Rectangular prism (cuboid) | Rectangle | l×w×h | 6 (rectangles) |
| Pentagonal prism | Pentagon | A_{\text{pent}} \cdot L | 7 |
| Hexagonal prism | Hexagon | A_{\text{hex}} \cdot L | 8 |
Pyramids (one polygonal base, triangular lateral faces meeting at an apex)
| Pyramid | Base shape | Volume | Faces |
|---|---|---|---|
| Triangular pyramid (tetrahedron) | Triangle | (\frac{1}{3} A_{\text{base}} h) | 4 (triangles) |
| Rectangular pyramid | Rectangle | (\frac{1}{3} lwh) | 5 |
| Pentagonal pyramid | Pentagon | (\frac{1}{3} A_{\text{pent}} h) | 6 |
| Hexagonal pyramid | Hexagon | (\frac{1}{3} A_{\text{hex}} h) | 7 |
The five Platonic solids
| Solid | Faces | Vertices | Edges | Face shape |
|---|---|---|---|---|
| Tetrahedron | 4 | 4 | 6 | Equilateral triangle |
| Cube | 6 | 8 | 12 | Square |
| Octahedron | 8 | 6 | 12 | Equilateral triangle |
| Dodecahedron | 12 | 20 | 30 | Regular pentagon |
| Icosahedron | 20 | 12 | 30 | Equilateral triangle |
Euler's formula holds for all five (and every convex polyhedron): V−E+F=2.
Curved solids
| Solid | Faces | Volume formula | Surface area formula |
|---|---|---|---|
| Sphere | One curved surface | (\frac{4}{3}\pi r^3) | 4πr² |
| Cylinder | Two circular bases + curved side | πr²h | 2πr(r+h) |
| Cone | One circular base + curved side | (\frac{1}{3}\pi r^2 h) | πr(r+ℓ) |
| Hemisphere | Half a sphere | (\frac{2}{3}\pi r^3) | 3πr² (curved + flat) |
| Torus | Donut shape | 2π²Rr² | 4π²Rr |
Summary of Geometry Taxonomy
- Geometric shapes split into 2D (length and width) and 3D (length, width, and height).
- 2D shapes group into polygons and non-polygons.
- 3D shapes group into polyhedra and curved solids.
- Euler's formula V−E+F=2 holds for every convex polyhedron.