Volume Notes

1. Volume is about three-dimensional space: how much space an object occupies.
2. In IB MYP/Standard Mathematics, volume connects geometry (solids), unit conversion, and real applications such as capacity.

Volume Units Work Differently From Length Units

1. For length, each step (for example from m to cm) is a factor of 100.
2. For area, each step is squared.
3. For volume, each step is cubed.
4. That single idea explains most volume conversions.

Why Squared And Cubed Units Grow So Fast

1. If you scale a shape by a factor of $k$:
   1. Lengths scale by $k$
   2. Areas scale by $k^2$
   3. Volumes scale by $k^3$
2. So changing from meters to centimeters multiplies lengths by $100$, but multiplies volumes by $100^3=1{,}000{,}000$.

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$^3$

1. Since $1\text{ m}=100\text{ cm}$,
   1. a cube of side $1\text{ m}$ has volume $1\text{ m}^3$
   2. a cube of side $100\text{ cm}$ has volume $(100\text{ cm})^3=1{,}000{,}000\text{ cm}^3$
2. So, $$1\text{ m}^3=1{,}000{,}000\text{ cm}^3$$

Converting cm$^3$ And mm$^3$

1. Since $1\text{ cm}=10\text{ mm}$, $$1\text{ cm}^3=(10\text{ mm})^3=1000\text{ mm}^3.$$
2. So:
   1. cm$^3$ to mm$^3$: multiply by $1000$
   2. mm$^3$ to cm$^3$: divide by $1000$

Volume And Capacity: Connecting cm³, ml, And Liters

1. In everyday measurement, capacity is often measured in liters.
2. A key equivalence is: $$1\text{ cm}^3=1\text{ ml}.$$
3. That means: $$1000\text{ cm}^3=1000\text{ ml}=1\text{ L}.$$

Core Volume Formulas For Common Solids

Volume is always "area of a cross-section/base times a suitable height", sometimes with a scale factor.

1. Cuboid (rectangular prism)
   1. A cuboid has length $l$, width $w$, height $h$: $$V=lwh$$
   2. Equivalently: $$V=(\text{area of base})\times h$$
2. Cylinder
   1. A cylinder has radius $r$ and height $h$.
   2. The base is a circle of area $\pi r^2$: $$V=\pi r^2h$$
3. Pyramid
   1. A pyramid is one-third of a prism with the same base area $B$ and height $h$: $$V=\frac13Bh$$
4. Cone
   1. A cone is one-third of a cylinder with the same base radius and height: $$V=\frac13\pi r^2h$$

The Volume Of A Sphere From A Circumscribed Cylinder

1. A sphere is the set of points at distance $r$ from a center. Its volume formula is: $$V=\frac43\pi r^3$$
2. One classic way to understand this result comes from comparing a sphere to the smallest cylinder that just fits around it (a circumscribed cylinder):
   1. cylinder radius $=r$
   2. cylinder height $=2r$
3. The cylinder's volume is: $$V_{\text{cyl}}=\pi r^2(2r)=2\pi r^3$$
4. Explorations (attributed historically to Archimedes) show that the sphere's volume is two-thirds of this circumscribed cylinder: $$V_{\text{sphere}}=\frac{2}{3}V_{\text{cyl}}=\frac{2}{3}(2\pi r^3)=\frac{4}{3}\pi r^3$$

Measuring Volume By Water Displacement

1. Not all objects have a simple formula.
2. If a solid sinks in water (for example a golf ball), you can find its volume using a measuring cylinder.
3. Method (displacement):
   1. Put the object in the cylinder and add water until it covers the object.
   2. Record the water level (object + water).
   3. Remove the object and record the new water level (water only).
   4. The difference is the object's volume.
4. Because $1\text{ ml}=1\text{ cm}^3$, the difference in ml is directly the volume in cm$^3$.

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

1. Since $1\text{ L}=1000\text{ cm}^3$, you need a cuboid with volume $1000\text{ cm}^3$.
2. Any dimensions with product 1000 will work, for example:
   1. $10\text{ cm}\times 10\text{ cm}\times 10\text{ cm}$ (a cube)
   2. $20\text{ cm}\times 10\text{ cm}\times 5\text{ cm}$
3. 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)

1. For a rectangular pool, approximate volume as a cuboid: $$V=\text{length}\times\text{width}\times\text{average depth}$$
2. Then convert m$^3$ to liters using $1\text{ m}^3=1000\text{ L}$.