3D Shapes — Volume (AQA GCSE Maths): Revision Notes

3D shapes — Volume

Understanding volume in 3D shapes

Volume tells us how much space a 3D shape takes up, measured in cubic units like cm³ or m³. Different 3D shapes have different volume formulas, but they all follow logical patterns once you understand the underlying concepts.

Volumes of prisms

A prism is a 3D shape that maintains the same cross-sectional area all the way through its length. Think of it like a loaf of bread - every slice would be identical in shape and size.

The key formula for any prism is: $V = A \times L$

Where:

• $V$ = Volume
• $A$ = Cross-sectional Area
• $L$ = Length

This works because you're essentially stacking identical flat shapes on top of each other to create the 3D object.

Triangular prisms

For a triangular prism, you first calculate the area of the triangular cross-section, then multiply by the length of the prism.

Cylinders

A cylinder is actually a special type of prism where the cross-section is a circle. The cross-sectional area is $\pi r^2$ (the area of a circle), so the volume becomes:

$V = \pi r^2 h$

where $r$ is the radius and $h$ is the height.

Practical example with cylinders

Let's look at a real-world application using density and mass calculations.

When working with cylinders, you can calculate not just volume but also mass if you know the density. The relationship is: $\text{mass} = \text{density} \times \text{volume}$

Volumes of spheres

A sphere is a perfectly round 3D shape where every point on the surface is the same distance from the centre.

$\text{Volume of sphere} = \frac{4}{3}\pi r^3$

Hemispheres

A hemisphere is exactly half a sphere, so its volume is simply half the sphere's volume: $\text{Volume of hemisphere} = \frac{2}{3}\pi r^3$

Volumes of pyramids and cones

These shapes start with a base and taper to a single point at the top. The key insight is that all pyramids and cones have a volume that's one-third of what a prism or cylinder with the same base and height would have.

$\text{Volume of pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Vertical Height}$

$\text{Volume of cone} = \frac{1}{3} \times \pi r^2 \times h$

Important height distinction

Volumes of frustums

A frustum is what you get when you "chop off" the top of a cone parallel to its base. Think of a bucket or lampshade shape.

To find the volume of a frustum: $\text{Volume of frustum} = \text{Volume of original cone} - \text{Volume of removed cone}$

When the cones are similar (same shape, different size), you can use scale factors to find the dimensions of the removed cone.

Rates of flow

Volume calculations become practical when dealing with rates of flow - how quickly liquid flows into or out of containers.

Key formulas and relationships