Geometry Basics

Geometry

Definition

Geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects.

Key Concepts

• Points: A location in space, represented by a set of coordinates (x, y, z).
• Lines: A set of points extending infinitely in two directions.
• Rays: A line that extends infinitely in one direction from a point.
• Angles: Formed by two rays sharing a common endpoint (vertex).

Properties of 2D Shapes

• Triangles:
  • Angles sum to 180°
  • Sum of any two sides is greater than the third side
• Quadrilaterals:
  • Sum of interior angles is 360°
  • Diagonals bisect each other (in rectangles and squares)
• Circles:
  • Center: fixed point equidistant from all points on the circle
  • Radius: distance from center to any point on the circle
  • Circumference: distance around the circle

Properties of 3D Shapes

• Pyramids:
  • Apex: highest point
  • Base: polygonal face
• Prisms:
  • Two identical polygonal faces (bases)
  • Rectangular sides
• Spheres:
  • Center: fixed point equidistant from all points on the sphere
  • Radius: distance from center to any point on the sphere
  • Surface area: 4 × π × radius²

Theorems and Formulas

• Pythagorean Theorem: a² + b² = c² (right-angled triangles)
• Distance Formula: √((x2 - x1)² + (y2 - y1)²) (distance between two points)
• Midpoint Formula: ((x1 + x2)/2, (y1 + y2)/2) (midpoint of a line segment)