- Volume is about three-dimensional space: how much space an object occupies.
- In IB MYP/Standard Mathematics, volume connects geometry (solids), unit conversion, and real applications such as capacity.
Volume
The amount of space a 3-dimensional solid occupies, measured in cubic units such as cm$^3$.
Capacity
The amount of space available to hold something, often measured in liters (L) or milliliters (ml).
Volume Units Work Differently From Length Units
- For length, each step (for example from m to cm) is a factor of 100.
- For area, each step is squared.
- For volume, each step is cubed.
- That single idea explains most volume conversions.
Why Squared And Cubed Units Grow So Fast
- If you scale a shape by a factor of $k$:
- Lengths scale by $k$
- Areas scale by $k^2$
- Volumes scale by $k^3$
- So changing from meters to centimeters multiplies lengths by $100$, but multiplies volumes by $100^3=1{,}000{,}000$.
- A very common mistake is to convert m$^3$ to cm$^3$ by multiplying by 100 or 10,000.
- For volume you must cube the linear conversion factor.
Cubes Make Unit Conversion Visual
A cube is the easiest solid for understanding volume because its volume is side$^3$.
Converting m$^3$ And cm$^3$
- Since $1\text{ m}=100\text{ cm}$,
- a cube of side $1\text{ m}$ has volume $1\text{ m}^3$
- a cube of side $100\text{ cm}$ has volume $(100\text{ cm})^3=1{,}000{,}000\text{ cm}^3$
- So, $$1\text{ m}^3=1{,}000{,}000\text{ cm}^3$$
Converting cm$^3$ And mm$^3$
- Since $1\text{ cm}=10\text{ mm}$, $$1\text{ cm}^3=(10\text{ mm})^3=1000\text{ mm}^3.$$
- So:
- cm$^3$ to mm$^3$: multiply by $1000$
- mm$^3$ to cm$^3$: divide by $1000$
- When converting cubic units, find the linear factor first, then cube it.
- For instance, cm to mm is $\times 10$, so cm$^3$ to mm$^3$ is $\times 10^3=\times 1000$.
Volume And Capacity: Connecting cm³, ml, And Liters
- In everyday measurement, capacity is often measured in liters.
- A key equivalence is: $$1\text{ cm}^3=1\text{ ml}.$$
- That means: $$1000\text{ cm}^3=1000\text{ ml}=1\text{ L}.$$
- cm$^3$ to ml: the number stays the same
- ml to cm$^3$: the number stays the same
- cm$^3$ to L: divide by $1000$
- L to cm$^3$: multiply by $1000$
- You will often measure volume of a solid in cm$^3$, but talk about capacity of containers in ml or liters.
- The units link directly through $1\text{ cm}^3=1\text{ ml}$.
Core Volume Formulas For Common Solids
Volume is always "area of a cross-section/base times a suitable height", sometimes with a scale factor.
- Cuboid (rectangular prism)
- A cuboid has length $l$, width $w$, height $h$: $$V=lwh$$
- Equivalently: $$V=(\text{area of base})\times h$$
- Cylinder
- A cylinder has radius $r$ and height $h$.
- The base is a circle of area $\pi r^2$: $$V=\pi r^2h$$
- Pyramid
- A pyramid is one-third of a prism with the same base area $B$ and height $h$: $$V=\frac13Bh$$
- Cone
- A cone is one-third of a cylinder with the same base radius and height: $$V=\frac13\pi r^2h$$
- Before substituting values, write the formula with symbols first (for example $V=\pi r^2h$).
- Then check: do your units match (cm with cm, not m with cm)?
- Only then calculate.
The Volume Of A Sphere From A Circumscribed Cylinder
- A sphere is the set of points at distance $r$ from a center. Its volume formula is: $$V=\frac43\pi r^3$$
- One classic way to understand this result comes from comparing a sphere to the smallest cylinder that just fits around it (a circumscribed cylinder):
- cylinder radius $=r$
- cylinder height $=2r$
- The cylinder's volume is: $$V_{\text{cyl}}=\pi r^2(2r)=2\pi r^3$$
- Explorations (attributed historically to Archimedes) show that the sphere's volume is two-thirds of this circumscribed cylinder: $$V_{\text{sphere}}=\frac{2}{3}V_{\text{cyl}}=\frac{2}{3}(2\pi r^3)=\frac{4}{3}\pi r^3$$
- Think of the circumscribed cylinder as the "tightest can" that can hold the sphere.
- The sphere does not fill the can completely, it uses exactly two-thirds of the can's volume when the can's height is the sphere's diameter.
Measuring Volume By Water Displacement
- Not all objects have a simple formula.
- If a solid sinks in water (for example a golf ball), you can find its volume using a measuring cylinder.
- Method (displacement):
- Put the object in the cylinder and add water until it covers the object.
- Record the water level (object + water).
- Remove the object and record the new water level (water only).
- The difference is the object's volume.
- Because $1\text{ ml}=1\text{ cm}^3$, the difference in ml is directly the volume in cm$^3$.
If the cylinder reads 86 ml with the ball and 44 ml without the ball, then the ball's volume is $86-44=42 \text{ ml} =42 \text{ cm}^3$.
Worked Conversions You Must Be Able To Do
Converting cm² To mm² (Area Reminder)
Because $1\text{ cm}=10\text{ mm}$,
$$1\text{ cm}^2=(10\text{ mm})^2=100\text{ mm}^2$$
So $20\text{ cm}^2=20\times 100=2000\text{ mm}^2$
Converting mm³ To cm³
Since $1\text{ cm}^3=1000\text{ mm}^3$,
$$5000\text{ mm}^3=\frac{5000}{1000}=5\text{ cm}^3$$
Converting m³ To Liters
Use $1\text{ m}^3=1{,}000{,}000\text{ cm}^3$ and $1000\text{ cm}^3=1\text{ L}$.
So,
$$1\text{ m}^3=\frac{1{,}000{,}000}{1000}\text{ L}=1000\text{ L}$$
Therefore $10\text{ m}^3=10{,}000\text{ L}$.
- Do not confuse $1\text{ m}^3$ with $1$ liter.
- A cubic meter is very large: it equals $1000$ liters.
Multi-Step Problems: Dimensions, Capacity, And Real Contexts
A typical modelling task is designing a container with a given capacity.
Designing A 1-Liter Carton
- Since $1\text{ L}=1000\text{ cm}^3$, you need a cuboid with volume $1000\text{ cm}^3$.
- Any dimensions with product 1000 will work, for example:
- $10\text{ cm}\times 10\text{ cm}\times 10\text{ cm}$ (a cube)
- $20\text{ cm}\times 10\text{ cm}\times 5\text{ cm}$
- In real packaging you would also consider practical constraints (stability, shape for stacking), but mathematically the key condition is $lwh=1000$.
Swimming Pool Capacity (Typical Setup)
- For a rectangular pool, approximate volume as a cuboid: $$V=\text{length}\times\text{width}\times\text{average depth}$$
- Then convert m$^3$ to liters using $1\text{ m}^3=1000\text{ L}$.
- What is the multiplier from cm$^3$ to mm$^3$?
- A cone and cylinder have the same $r$ and $h$. How do their volumes compare?