Volume and Density — Quick Reference

Volume

  • Volume: amount of space an object occupies.
  • SI unit: m3\text{m}^3; derived from base length.
  • A cube with edge length 1 m has volume V=13=1 m3V = 1^3 = 1\ \text{m}^3.
  • Common practical units:
    • cubic decimeter (dm$^3$) and liter (L): 1 dm3=1 L1\ \text{dm}^3 = 1\ \text{L}; 1 L ≈ 1.06 qt.
    • cubic centimeter (cm$^3$) = 1 mL; 1 cm$^3$ = 1 mL; cc is often used in health fields.
    • 1 m$^3$ = 1000 L (and = 1\,000,000 cm$^3$).
  • Volume of a cube: V=a3V = a^3 for edge length aa.
  • 1 L is the cubic decimeter; 1 cm$^3$ is the volume of a cube 1 cm on each side.

Density

  • Density is the ratio of mass to volume: D=MVD = \frac{M}{V}.
  • SI unit: kg/m3\text{kg/m}^3; common practical units: g/cm3, g/mL, g/L\text{g/cm}^3,\ g/mL,\ g/L.
  • Conversion notes:
    • 1 g/cm3=1000 kg/m31\ \text{g/cm}^3 = 1000\ \text{kg/m}^3; 1 g/mL=1 g/cm31\ \text{g/mL} = 1\ \text{g/cm}^3.
  • Mass-Volume relationship:
    • M=D⋅VM = D \cdot V
    • V=MDV = \frac{M}{D}
  • How to determine density:
    • Measure mass and volume, then compute D=MVD = \dfrac{M}{V}.
    • For irregular objects, use water displacement to find volume: V<em>extobject=V</em>extfinal−VextinitialV<em>{ ext{object}} = V</em>{ ext{final}} - V_{ ext{initial}}.

Example: lead cube

  • Given: edge length = 2.00 cm2.00\ \text{cm}, mass = 90.7 g90.7\ \text{g}.
  • Volume: V=(2.00 cm)3=8.00 cm3V = (2.00\ \text{cm})^3 = 8.00\ \text{cm}^3.
  • Density: D=90.7 g8.00 cm3=11.3 g/cm3D = \dfrac{90.7\ \text{g}}{8.00\ \text{cm}^3} = 11.3\ \text{g/cm}^3.

Example: irregular solid (rubber stopper)

  • Mass = 10.32 g10.32\ \text{g}.
  • Water displacement: initial = 30.0 mL30.0\ \text{mL}, final = 36.0 mL36.0\ \text{mL}.
  • Volume of object: V=36.0−30.0=6.0 mLV = 36.0 - 30.0 = 6.0\ \text{mL}.
  • Density: D=10.32 g6.0 mL=1.7 g/mLD = \dfrac{10.32\ \text{g}}{6.0\ \text{mL}} = 1.7\ \text{g/mL}.

Quick reference: common densities (selected)

  • Water: 1.0 g/cm31.0\ \text{g/cm}^3
  • Ice: 0.92 g/cm30.92\ \text{g/cm}^3
  • Oak: 0.60–0.90 g/cm30.60 \text{–} 0.90\ \text{g/cm}^3
  • Copper: ≈9.0 g/cm3\approx 9.0\ \text{g/cm}^3
  • Gold: 19.3 g/cm319.3\ \text{g/cm}^3
  • Ethanol: 0.79 g/cm30.79\ \text{g/cm}^3
  • Glycerin: 1.26 g/cm31.26\ \text{g/cm}^3
  • Mercury: 13.6 g/cm313.6\ \text{g/cm}^3
  • Gas densities (air-like): Helium 0.16 g/L0.16\ \text{g/L}, Neon 0.83 g/L0.83\ \text{g/L}, Dry air 1.20 g/L1.20\ \text{g/L}

Summary formulas

  • Density: D=MVD = \dfrac{M}{V}
  • Mass from density and volume: M=D⋅VM = D\cdot V
  • Volume from mass and density: V=MDV = \dfrac{M}{D}
  • Irregular volume (displacement): V<em>extobject=V</em>extfinal−VextinitialV<em>{ ext{object}} = V</em>{ ext{final}} - V_{ ext{initial}}