Geometrical Shapes in Math

Geometrical Shapes in Math

Geometrical shapes in math are figures defined by points, lines, angles, and surfaces. They can be classified based on dimensions (2D or 3D), regularity, and properties. Here’s a structured breakdown:

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### **1. 2D (Plane) Shapes**  

Flat shapes with **length** and **width** (no depth).  

#### **Basic Polygons** (Closed shapes with straight sides):  

| Shape          | Sides/Angles | Properties |  

|----------------|-------------|------------|  

| **Triangle**   | 3           | Sum of angles = 180°; Types: Equilateral (all equal), Isosceles (2 equal), Scalene (all unequal), Right (1 angle = 90°). |  

| **Quadrilateral** | 4       | Sum of angles = 360°; Types: Square, Rectangle, Parallelogram, Rhombus, Trapezoid, Kite. |  

| **Pentagon**   | 5           | Regular: All sides/angles equal. |  

| **Hexagon**    | 6           | Common in tessellations. |  

| **Heptagon**   | 7           | Also called a septagon. |  

| **Octagon**    | 8           | Stop sign shape. |  

#### **Curved Shapes**:  

- **Circle**: All points equidistant from the center. Key terms: Radius, Diameter, Chord, Sector.  

- **Semicircle**: Half of a circle.  

- **Ellipse**: Stretched circle with two foci (e.g., planetary orbits).  

- **Oval**: No strict definition; often refers to egg-like curves.  

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### **2. 3D (Solid) Shapes**  

Have **length, width, and height**.  

#### **Polyhedrons** (Flat faces, straight edges):  

| Shape          | Faces/Edges/Vertices | Properties |  

|----------------|----------------------|------------|  

| **Cube**       | 6 squares, 12, 8     | All faces equal. |  

| **Cuboid**     | 6 rectangles, 12, 8  | Opposite faces equal. |  

| **Tetrahedron** | 4 triangles, 6, 4    | Simplest polyhedron. |  

| **Octahedron** | 8 triangles, 12, 6   | Like two pyramids joined at the base. |  

| **Dodecahedron** | 12 pentagons, 30, 20 | Plato’s "cosmic" shape. |  

| **Icosahedron** | 20 triangles, 30, 12 | Most faces among Platonic solids. |  

#### **Non-Polyhedrons** (Curved surfaces):  

- **Sphere**: All surface points equidistant from the center (e.g., a ball).  

- **Cylinder**: Two circular bases connected by a curved surface.  

- **Cone**: Circular base tapering to a point (apex).  

- **Torus**: Donut-shaped with a hole (e.g., a lifebuoy).  

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### **3. Special Categories**  

- **Regular vs. Irregular**: Regular shapes have equal sides/angles (e.g., equilateral triangle); irregular do not.  

- **Convex vs. Concave**: Convex shapes have no inward dents; concave shapes do (e.g., a crescent moon).  

- **Symmetry**:  

  - **Line Symmetry**: Shapes divisible into mirror images (e.g., a heart).  

  - **Rotational Symmetry**: Shapes look identical when rotated (e.g., a starfish).  

- **Fractals**: Infinitely complex patterns (e.g., Koch snowflake, Mandelbrot set).  

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### **Key Formulas**  

#### **2D Shapes**:  

- **Area**:  

  - Circle: \( \pi r^2 \)  

  - Triangle: \( \frac{1}{2} \times \text{base} \times \text{height} \)  

  - Rectangle: \( \text{length} \times \text{width} \)  

- **Perimeter**: Sum of all sides.  

#### **3D Shapes**:  

- **Volume**:  

  - Cube: \( \text{side}^3 \)  

  - Sphere: \( \frac{4}{3}\pi r^3 \)  

  - Cylinder: \( \pi r^2 h \)  

- **Surface Area**: Sum of all face areas.  

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### **Visual Examples**  

- **2D**: Squares, circles, pentagons.  

- **3D**: Cubes, pyramids, spheres.  

- **Advanced**: Hyperboloids (3D curves), Tesseracts (4D cubes).  

Need details on a specific shape or its properties? Let me know!