# 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.