🔺 Geometry Basics (Shapes, Perimeter & Area)

Last Updated: 06-Sep-2025

Geometry is the study of shapes, sizes, and space.

📌 Basic 2D Shapes

  • Triangle → 3 sides
  • Square → 4 equal sides
  • Rectangle → Opposite sides equal
  • Circle → Round, no sides

👉 Real Life:

  • Triangle → Traffic sign
  • Rectangle → Book, phone screen
  • Circle → Coin, wheel

Perimeter (Boundary Length)

Perimeter = distance around a shape.

  • Square → 4 × side
  • Rectangle → 2 × (length + width)
  • Triangle → sum of 3 sides
  • Circle → 2πr (r = radius)

👉 Example:
Square side = 5 cm → Perimeter = 4×5 = 20 cm.

Area (Space inside)

Area = region covered inside shape.

  • Square → side²
  • Rectangle → length × width
  • Triangle → ½ × base × height
  • Circle → πr²

👉 Example:
Circle radius = 7 cm → Area ≈ 3.14 × 7² = 153.86 cm².

Angles

  • Right angle = 90°
  • Acute angle < 90°
  • Obtuse angle > 90° but < 180°
  • Straight angle = 180°

👉 Real Life:

  • 90° → Corner of a paper.
  • 180° → Flat line.

3D Shapes (Solid Geometry)

  • Cube → Box with all equal sides
  • Cuboid → Box (like brick)
  • Sphere → Ball
  • Cylinder → Pipe
  • Cone → Ice-cream cone

👉 Extra formulas:

  • Volume of cube = side³
  • Volume of cuboid = l × w × h
  • Volume of sphere = 4/3 πr³

Pythagoras Theorem

In a right-angled triangle,
(Hypotenuse)² = (Base)² + (Height)²

👉 Example:

  • Base = 3, Height = 4 → Hypotenuse = √(3² + 4²) = √25 = 5

Polygons

A polygon is a closed shape made of straight lines.

  • Triangle → 3 sides
  • Quadrilateral → 4 sides
  • Pentagon → 5 sides
  • Hexagon → 6 sides

👉 Formula:

  • Sum of interior angles = (n − 2) × 180°
    • Triangle → (3−2)×180 = 180°
    • Pentagon → (5−2)×180 = 540°

Circles

Parts of a circle:

  • Radius (r): center to boundary
  • Diameter (d): 2 × radius
  • Circumference (C): distance around circle = 2πr
  • Chord: line joining two points on circle
  • Arc: part of boundary
  • Sector: slice of a circle (like pizza piece)

👉 Example:
Circle radius = 7 → Diameter = 14, Circumference ≈ 44.

Symmetry

  • A shape has line of symmetry if it can be folded into two equal halves.
  • Example:
    • Square → 4 symmetry lines
    • Rectangle → 2 symmetry lines
    • Circle → infinite symmetry lines

Coordinate Geometry Basics

We use an X-axis (horizontal) and Y-axis (vertical) to locate points.

  • A point is written as (x, y).
  • Example:
    • (2,3) → move 2 right on X, 3 up on Y.
  • Distance between (x₁,y₁) and (x₂,y₂):
    √((x₂−x₁)² + (y₂−y₁)²)

👉 Example:
Distance between (0,0) and (3,4) = √(3²+4²) = 5 (again Pythagoras!).

⚡ Quick Practice

  1. A right triangle has base 5, height 12. Find hypotenuse.
  2. Find sum of angles in a hexagon.
  3. Circle radius = 14. Find diameter and circumference.
  4. Draw a square and mark all lines of symmetry.
  5. Find distance between (1,2) and (4,6).
  6. Find perimeter of rectangle (l=8 cm, w=5 cm).
  7. Find area of triangle (base=6 cm, height=4 cm).
  8. A circle has radius 10 cm. Find its area and circumference.
  9. Identify angle type: 60°, 120°, 180°.
  10. A cube has side 3 cm. Find its volume.
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