Surface Areas and Volumes

• Introduction & Basic Solids

  • Cuboid
    • Total Surface Area (TSA): $2(lb + bh + hl)$
    • Curved/Lateral Surface Area (LSA): $2(l + b)h$
    • Volume: $l \times b \times h$
    • Diagonal: $\sqrt{l^2 + b^2 + h^2}$
  • Cube
    • TSA: $6a^2$
    • LSA: $4a^2$
    • Volume: $a^3$
  • Cylinder
    • Curved Surface Area (CSA): $2\pi rh$
    • TSA: $2\pi r(r + h)$
    • Volume: $\pi r^2h$
  • Cone
    • Slant Height ($l$): $\sqrt{r^2 + h^2}$
    • CSA: $\pi rl$
    • TSA: $\pi r(r + l)$
    • Volume: $\frac{1}{3}\pi r^2h$
  • Sphere
    • Surface Area: $4\pi r^2$
    • Volume: $\frac{4}{3}\pi r^3$
  • Hemisphere
    • CSA: $2\pi r^2$
    • TSA: $3\pi r^2$
    • Volume: $\frac{2}{3}\pi r^3$

• Surface Area of Combination of Solids

  • Concept: Total area is the sum of visible surface areas of individual solids.
  • Excluded Regions: Areas of contact between joined solids are subtracted.
  • Examples
    • Toy made of hemisphere and cone
    • Tent with cylindrical base and conical top
    • Capsule (cylinder with two hemispherical ends)

• Volume of Combination of Solids

  • Concept: Total volume is the sum of volumes of individual constituent parts.
  • Examples
    • Solid iron pole (cylinder topped by cylinder)
    • Vessel in form of hollow hemisphere mounted by hollow cylinder

• Conversion of Solids (Recasting)

  • Core Principle: Conservation of Volume
  • Formula: $\text{Volume of Initial Solid} = \text{Volume of New Solid(s)}$
  • Examples
    • Melting a sphere into metallic cones/cylinders
    • Recasting a solid metallic cylinder into wire
    • Embankment soil calculation from well digging

• Frustum of a Cone

  • Definition: Cut a right circular cone by a plane parallel to its base.
  • Dimensions: Radii of ends ($r_1, r_2$), Height ($h$), Slant height ($l$)
  • Formulas
    • Slant Height ($l$): $\sqrt{h^2 + (r_1 - r_2)^2}$
    • CSA: $\pi(r_1 + r_2)l$
    • TSA: $\pi(r_1 + r_2)l + \pi r_1^2 + \pi r_2^2$
    • Volume: $\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1r_2)$
  • Real-world Examples: Buckets, drinking glasses, fez caps