Volume & Surface Area

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Compute volume and surface area of common 3D shapes.
  • Match formulas to appropriate solids.
  • Interpret word problems involving capacity, wrapping, and covering surfaces.
  • Understand how scaling affects volume and surface area.

Key Ideas

Volume: measures space inside a solid
Surface area: measures total area of all faces

Formulas

Prisms & Cylinders (Base Area × Height)

  • Volume:
    \[ V = Bh \]
  • Surface Area (prism): add areas of all faces
  • Cylinder surface area:
    \[ SA = 2\pi r^2 + 2\pi rh \]

Pyramids & Cones

  • Volume:
    \[ V = \frac{1}{3}Bh \]
  • Surface Area: sum of base area + triangular/lateral faces

Sphere

  • Volume:
    \[ V = \frac{4}{3}\pi r^3 \]
  • Surface Area:
    \[ SA = 4\pi r^2 \]
Shape Volume Surface Area
Rectangular Prism \(lwh\) \(2(lw + lh + wh)\)
Cylinder \(\pi r^2 h\) \(2\pi r^2 + 2\pi rh\)
Pyramid \(\frac13Bh\) base + lateral faces
Cone \(\frac13\pi r^2 h\) \(\pi r^2 + \pi r\ell\)
Sphere \(\frac43\pi r^3\) \(4\pi r^2\)

Scaling

If scale factor = \(k\):

  • Surface area multiplies by \(k^2\)
  • Volume multiplies by \(k^3\)

Scaling comparison showing how surface area grows by \(k^2\) and volume grows by \(k^3\).

Common Problem Types

Compute Volume Directly

Example: Cylinder with \(r=3\), \(h=10\):
\(V = \pi (3)^2 (10) = 90\pi\).

Compute Surface Area

Example: Rectangular prism → add all face areas.

Word Problems (Capacity, Filling, Packing)

Example: How many cubic inches can a box hold?

Composite Solids

Break shapes into parts.
Example: A dome on top of a cylinder → add cylinder + hemisphere volumes.

Scaling Questions

If radius doubles:
- surface area ×4
- volume ×8

Nets for Surface Area

Use net to compute total area of faces.

Strategies

  • Identify the base area first.
  • Label dimensions clearly before plugging into formulas.
  • For composite solids: compute each piece separately.
  • Check that units match (don’t mix cm and m).
  • Use \(\pi\) symbol whenever the problem allows.

Worked Examples

Example 1 — Volume of a Cylinder

\(r=4\), \(h=12\)
\[ V = \pi \cdot 4^2 \cdot 12 = 192\pi \]

Example 2 — Surface Area of a Rectangular Prism

Dimensions 3 × 4 × 5
\[ SA = 2(3\cdot4 + 3\cdot5 + 4\cdot5) = 2(12 + 15 + 20) = 94 \]

Example 3 — Composite Solid

Cylinder radius 2, height 6, topped with hemisphere radius 2.
\[ V = \pi(2^2)(6) + \frac{2}{3}\pi (2^3) = 24\pi + \frac{16}{3}\pi \]

WarningCommon Mistakes
  • Mixing up area and volume formulas.
  • Forgetting the \(\frac13\) factor for pyramids/cones.
  • Not including both circular areas in cylinder surface area.
  • Confusing slant height \(\ell\) with height \(h\) in cones.
  • Adding dimensions instead of multiplying base area.

Practice Problems

  1. Volume of a rectangular prism 4 × 5 × 6
  2. Volume of cone: \(r=3\), \(h=9\)
  3. Surface area of cylinder: \(r=2\), \(h=10\)
  4. If a cube’s edge doubles, how do volume and surface area change?
  1. \(4\cdot5\cdot6 = 120\)
  2. \(\frac13\pi (3^2)(9) = 27\pi\)
  3. \(SA = 2\pi r^2 + 2\pi rh = 8\pi + 40\pi = 48\pi\)
  4. SA ×4, Volume ×8

Summary

  • Volume measures space; surface area measures covering.
  • Prisms/cylinders use \(Bh\); pyramids/cones use \(\frac13Bh\).
  • Spheres: \(V=\frac43\pi r^3\), \(SA=4\pi r^2\).
  • Scaling affects area and volume differently.
  • Identify the base first.
  • Use nets for surface area.
  • Keep \(\pi\) symbol unless decimal is required.
  • Composite shapes → calculate each part separately.