Volume (Edexcel GCSE Maths): Revision Notes

Volume

Volume is a fundamental concept in geometry that measures the amount of space a three-dimensional object occupies. You might also hear it called capacity in some contexts. Understanding how to calculate volume is essential for solving real-world problems involving containers, construction materials, and many other practical applications.

Volumes of cuboids

A cuboid is essentially a rectangular block - a 3D shape with six rectangular faces. Calculating its volume is straightforward and forms the foundation for understanding more complex volume calculations.

The volume of a cuboid equals the product of its three dimensions. When you multiply length, width, and height together, you get the total space inside the shape.

Formula: $V = L \times W \times H$

Volumes of prisms

A prism is a special type of 3D shape that maintains the same cross-sectional area throughout its entire length. Think of it like a loaf of bread - every slice would have exactly the same shape and size.

The key principle for all prisms is that volume equals the cross-sectional area multiplied by the length of the prism.

Formula: $V = A \times L$

Where $A$ is the area of the cross-section and $L$ is the length.

Triangular prisms

For triangular prisms, you first need to calculate the area of the triangular cross-section using the standard triangle area formula ($\frac{1}{2} \times \text{base} \times \text{height}$), then multiply by the length of the prism.

Cylinders

A cylinder is actually a special type of prism with a circular cross-section. Since the cross-sectional area of a circle is $\pi r^2$, the volume formula becomes:

Formula: $V = \pi r^2 h$

Where $r$ is the radius of the circular base and $h$ is the height of the cylinder.

Volumes of spheres

A sphere is a perfectly round 3D shape where every point on the surface is equidistant from the centre. The volume formula for a sphere is more complex than other shapes:

Formula: $V = \frac{4}{3}\pi r^3$

A hemisphere is exactly half a sphere, so its volume is simply half the sphere formula:

Formula for hemisphere: $V = \frac{2}{3}\pi r^3$

Volumes of pyramids and cones

Pyramids and cones are shapes that taper from a flat base to a single point at the top. The base can be any shape - if it's circular, we call it a cone; if it's any other shape, we call it a pyramid.

Formula for pyramid: $V = \frac{1}{3} \times \text{Base area} \times \text{Vertical height}$

Formula for cone: $V = \frac{1}{3} \times \pi r^2 \times h$

The key factor here is the "one-third" multiplier. This means that a pyramid or cone has exactly one-third the volume of a prism or cylinder with the same base area and height.

Volumes of frustums

A frustum is what remains when you cut off the top part of a cone parallel to its base. Think of a waste paper basket or a lampshade - these are common examples of frustums.

To find the volume of a frustum, you calculate it as the difference between two cones:

Formula: Volume of frustum = Volume of original cone - Volume of removed cone

This involves:

1. Finding the volume of the complete original cone
2. Finding the volume of the small cone that was removed from the top
3. Subtracting the second volume from the first