Mensuration (Class 8 Mathematics)

• 2D Figures (Plane Shapes)

  • Basic Concepts
    • Perimeter: Distance around the boundary
    • Area: Region enclosed inside the boundary
  • Standard Shapes
    • Square
      • Perimeter = $4 \times \mathbf{side}$
      • Area = $\mathbf{side}^2$
    • Rectangle
      • Perimeter = $2(\mathbf{length} + \mathbf{breadth})$
      • Area = $\mathbf{length} \times \mathbf{breadth}$
    • Triangle
      • Area = $\frac{1}{2} \times \mathbf{base} \times \mathbf{height}$
    • Parallelogram
      • Area = $\mathbf{base} \times \mathbf{height}$
    • Circle
      • Perimeter (Circumference) = $2\pi \mathbf{r}$
      • Area = $\pi \mathbf{r}^2$
  • Area of Special Quadrilaterals
    • Trapezium
      • Area = $\frac{1}{2} \times (\mathbf{a} + \mathbf{b}) \times \mathbf{h}$
      • $(\mathbf{a}, \mathbf{b})$ = parallel sides, $\mathbf{h}$ = perpendicular distance
    • Rhombus
      • Area = $\frac{1}{2} \times \mathbf{d_1} \times \mathbf{d_2}$
      • $\mathbf{d_1}, \mathbf{d_2}$ = lengths of diagonals
    • General Quadrilateral
      • Area = $\frac{1}{2} \times \mathbf{d} \times (\mathbf{h_1} + \mathbf{h_2})$
      • $\mathbf{d}$ = diagonal, $\mathbf{h_1}, \mathbf{h_2}$ = perpendicular heights from opposite vertices
  • Area of Polygons
    • Split polygon into non-overlapping triangles and trapeziums
    • Sum areas of all individual parts

• 3D Figures (Solid Shapes)

  • Cuboid
    • Total Surface Area (TSA) = $2(\mathbf{lb} + \mathbf{bh} + \mathbf{hl})$
    • Lateral Surface Area (LSA) = $2\mathbf{h}(\mathbf{l} + \mathbf{b})$
    • Volume = $\mathbf{l} \times \mathbf{b} \times \mathbf{h}$
  • Cube
    • Total Surface Area (TSA) = $6\mathbf{a}^2$
    • Lateral Surface Area (LSA) = $4\mathbf{a}^2$
    • Volume = $\mathbf{a}^3$
  • Right Circular Cylinder
    • Curved Surface Area (CSA) = $2\pi \mathbf{r}\mathbf{h}$
    • Total Surface Area (TSA) = $2\pi \mathbf{r}(\mathbf{r} + \mathbf{h})$
    • Volume = $\pi \mathbf{r}^2 \mathbf{h}$

• Volume and Capacity

  • Volume: Amount of space occupied by a 3D shape
  • Capacity: Quantity of liquid a container can hold
  • Key Unit Conversions
    • $1 \text{ cm}^3 = 1 \text{ mL}$
    • $1 \text{ L} = 1000 \text{ cm}^3$
    • $1 \text{ m}^3 = 1000 \text{ L}$