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Math101learn.math101.caVolume
Volume measures three-dimensional capacity in cubic units and connects base area, height, scale, and real containers.
Volume measures the amount of three-dimensional space inside a solid and is expressed in cubic units.
Why units are cubed
A cubic centimetre is a $1$ cm by $1$ cm by $1$ cm cube. A rectangular prism with dimensions $4$, $3$, and $2$ cm can be filled by $4\times3\times2=24$ such cubes, so its volume is $24$ cm³.
Volume describes capacity or occupied space; surface area describes the material covering the outside.
Prisms and cylinders
Any prism has constant cross-section, so
where $B$ is base area and $h$ is perpendicular length. For a rectangular prism, this becomes $V=lwh$.
A cylinder has circular base area $\pi r^2$:
Pyramids and cones
A pyramid or cone with the same base and perpendicular height as a prism or cylinder has one third the volume:
For a cone, $V=\tfrac13\pi r^2h$. Use vertical height, not slant height, because the formula measures perpendicular stacking from base to tip.
Spheres
A sphere of radius $r$ has volume
Because radius is cubed, a modest increase in radius produces a much larger change in volume. Doubling radius multiplies volume by $2^3=8$.
Composite solids
Split an object into non-overlapping familiar solids. Add pieces that are present and subtract cavities. A capsule shape, for example, may be modelled as a cylinder plus two hemispheres, which together form one sphere.
Sketch boundaries between components and keep a labelled expression before substituting numbers.
Missing dimensions
Rearrange formulas when volume is known. For a prism, $h=V/B$. If a rectangular tank has volume $360$ m³ and base dimensions $12$ m by $5$ m, then $B=60$ m² and $h=360/60=6$ m.
Check that the unknown dimension uses linear units, even though volume used cubic units.
Capacity conversions
One cubic centimetre equals one millilitre, and $1000$ cm³ equals one litre. One cubic metre equals $1000$ litres—not $100$—because each of three dimensions changes from centimetres to metres.
Convert dimensions before applying a volume formula whenever possible.
Scale factor
If all lengths scale by $k$, volume scales by $k^3$. A model built at half the original length in every direction has $(1/2)^3=1/8$ of the original volume.
Common mistakes
Using diameter as radius. Divide diameter by $2$ before squaring or cubing.
Forgetting the factor $1/3$. Pyramids and cones are one third of matching prisms or cylinders.
Using slant height. Volume uses perpendicular height.
Mixing units. Convert all dimensions to a common unit first.
Reporting square units. Volume requires cubic units.
Quick self-check
- What is the base, and what is its area?
- Is height perpendicular to the base?
- Is the solid prism-like, pyramid-like, spherical, or composite?
- Do the units and scale of the answer make sense?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the exact volume of a cylinder with radius 4 cm and height 10 cm?
- V = π(4)²(10)
- = π(16)(10)
- = 160π cm³
End of lesson
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