Geometric shapes are the spatial forms of objects, lines, and regions, described by mathematical principles.

They are broadly categorized into two-dimensional (2D) and three-dimensional (3D) shapes based on the number of dimensions they occupy.

These shapes have only length and width (or breadth) and no thickness. They are flat figures that can be drawn on a plane.

• Points: A location in space, with no dimension.
• Lines: A one-dimensional figure that extends infinitely in two directions.
• Angles: Formed by two rays sharing a common endpoint.
• Polygons: Closed figures made of straight line segments.

  • Triangles: Polygons with three sides (e.g., \( \text{equilateral, isosceles, scalene} \)).
  • Quadrilaterals: Polygons with four sides (e.g., \( \text{squares, rectangles, parallelograms, trapezoids} \)).
  • Pentagons, Hexagons, etc.: Polygons with five, six, or more sides.

• Circles: A set of all points in a plane that are equidistant from a central point.

These shapes have length, width, and height (or depth). They occupy space and have volume.

• Polyhedra: Solids with flat faces, straight edges, and sharp corners (vertices).

  • Cubes: Six square faces.
  • Prisms: Two identical bases and rectangular sides (e.g., \( \text{rectangular prism, triangular prism} \)).
  • Pyramids: A polygonal base and triangular faces that meet at an apex.

• Non-Polyhedra (Curved Solids):

  • Spheres: A perfectly round 3D object where every point on its surface is equidistant from its center.
  • Cylinders: Two parallel circular bases and a curved surface connecting them.
  • Cones: A circular base and a curved surface tapering to a single vertex (apex).
  • Torus: A donut-shaped surface generated by revolving a circle about an axis coplanar with the circle but not intersecting it.

Each shape has unique properties:

• Area/Surface Area: The measure of the surface covered by a 2D shape or the outer surface of a 3D shape.
• Perimeter/Circumference: The total distance around the boundary of a 2D shape.
• Volume: The amount of space occupied by a 3D shape.
• Vertices: Points where edges meet.
• Edges: Line segments where faces meet.
• Faces: Flat surfaces of a 3D shape.

Geometric shapes are essential in many fields:

• Architecture and Engineering: Design of buildings, bridges, and machines.
• Art and Design: Creating visual compositions and aesthetic structures.
• Physics: Understanding motion, forces, and spatial relationships.
• Computer Graphics: Modeling 3D objects and environments.
• Everyday Life: Recognizing objects, packing, and navigation.