Volume & Surface AreaEdexcel IGCSE Maths: Revision notes

A prism is any 3D solid with a constant cross-section along its length. This includes cuboids, triangular prisms and cylinders.

$V = \text{area of cross-section} \times \text{length}$

For a cuboid: $V = lwh$

For a cylinder: cross-section is a circle, so $V = \pi r^2 h$

Always identify the cross-section first, find its area, then multiply by the length/height running perpendicular to it.

Surface area is the total area of every face on the outside of the solid.

Method:

1. Identify all the faces (for a cuboid: 3 pairs of rectangles; for a cylinder: 2 circles + 1 curved rectangle).
2. Find the area of each face.
3. Add them all together.

For a cuboid: $SA = 2(lw + lh + wh)$

For a cylinder: $SA = 2\pi r^2 + 2\pi r h$ (two circular ends + curved surface, which unrolls into a rectangle of width $2\pi r$ and height $h$).

Pyramids and cones taper to a single apex, so their volume is one third of the equivalent prism/cylinder.

Pyramid volume: $V = \dfrac{1}{3} \times \text{base area} \times \text{height}$

Cone volume: $V = \dfrac{1}{3}\pi r^2 h$

Cone curved surface area: $SA_{\text{curved}} = \pi r l$, where $l$ is the slant height (not the vertical height).

Cone total surface area: $SA = \pi r l + \pi r^2$

The slant height, radius and vertical height form a right-angled triangle, so $l^2 = r^2 + h^2$ (Pythagoras) when you need to find one from the others.

A sphere has no flat faces or cross-section, so its formulae must simply be memorised (they are given on the Edexcel formula sheet).

Volume: $V = \dfrac{4}{3}\pi r^3$

Surface area: $SA = 4\pi r^2$

For a hemisphere (half a sphere), halve the sphere volume, but for surface area remember to add the flat circular face: $SA = 2\pi r^2 + \pi r^2 = 3\pi r^2$.

Composite solids are made of two or more basic solids joined together (e.g. a cylinder topped with a hemisphere, or a pyramid on top of a cuboid).

Method:

1. Split the solid into its basic parts.
2. Calculate volume/surface area of each part separately.
3. For volume: add the parts together.
4. For surface area: add the outer faces only — do NOT include any face that is now hidden inside the joined solid (e.g. the circular face where a hemisphere meets a cylinder is internal and excluded).

Always re-read the question to check whether it wants volume, surface area, or both, and give units cubed ($\text{cm}^3$) for volume and units squared ($\text{cm}^2$) for surface area.