Volume and Recasting of Solids: Quick Revision

A brief note on Volume and Recasting of Solids. Read it before a test. Try each question at the end before you open its answer.

The volumes

• Cuboid: $abc$. Cube: $a^3$. Cylinder: $\pi r^2 h$.
• Cone: $\dfrac{1}{3}\pi r^2 h$, with the vertical height $h$.
• Sphere: $\dfrac{4}{3}\pi r^3$. Hemisphere: $\dfrac{2}{3}\pi r^3$.
• A cone, a hemisphere and a cylinder with radius $r$ and height $r$ are in the ratio $1 : 2 : 3$.
• Volumes of joined parts add. A hollow is subtracted. One litre is $1000\ \text{cm}^3$.

Melting and flowing

• When a solid is melted and recast, the volume stays the same. The number of pieces is the old volume divided by the volume of one.
• $n$ equal balls made from one ball have $\sqrt[3]{n}$ times its surface area.
• Water from a pipe in time $t$ is $\pi r^2 v t$. The time to fill a tank is its volume divided by the volume per second.
• Earth from a well is $\pi r^2 d$. Spread over an area $A$, it has a height volume $\div A$.
• A solid under water raises the level by its volume divided by the area of the water surface.

A worked example

A solid cone with radius $3\ \text{cm}$ and height $8\ \text{cm}$ is melted and made into balls of radius $1\ \text{cm}$. How many balls are made?

1. The cone has volume $\dfrac{1}{3}\pi \times 9\ \text{cm}^2 \times 8\ \text{cm} = 24\pi\ \text{cm}^3$.
2. One ball has volume $\dfrac{4}{3}\pi\ \text{cm}^3$.
3. The number of balls is $24\pi \div \dfrac{4\pi}{3} = 18$.

Practice with new numbers

A program to try

The program finds the height of a cone that has the same volume as a ball of the same radius.

Mistakes to avoid

• Forgetting the $\frac{1}{3}$ of a cone or the $\frac{4}{3}$ of a ball.
• Equating areas when a solid is melted. The volume stays the same, not the area.
• Mixing units such as millimetres and centimetres, or litres and cubic metres.
• Using the diameter in the place of the radius.

Check yourself

1. A cylinder has radius $2\ \text{cm}$ and height $9\ \text{cm}$. What is its volume?

2. A ball of radius $6\ \text{cm}$ is melted into balls of radius $3\ \text{cm}$. How many?

3. Water leaves a pipe of cross-section $10\ \text{cm}^2$ at $30\ \text{cm/s}$. How much water leaves in one minute?

4. A ball of volume $100\ \text{cm}^3$ sinks in a jar of cross-section $50\ \text{cm}^2$. How much does the water rise?