Volume

A cuboid is a 3D solid with six rectangular faces (a box shape). To find the volume — the amount of space inside it — you multiply the three side lengths together: $V = l \times w \times h.$

Because you multiplied three lengths, your answer is in cubic units: $\text{cm}^3$, $\text{m}^3$ or $\text{mm}^3$.

A cube is just a cuboid with all three lengths equal, so $V = a^3$ where $a$ is the side length. A cube of side $5\,\text{cm}$ has volume $5^3 = 125\,\text{cm}^3$.

If a question gives the volume and two of the three lengths, you can find the missing length by dividing: $h = V \div (l \times w)$.

A prism is a 3D solid where the same cross-section runs all the way through. Slice through it anywhere along its length and you get the same shape each time. Triangular prisms, hexagonal prisms and even cuboids (a rectangular prism) all share one rule: $V = A \times l,$ where $A$ is the area of the cross-section and $l$ is the length of the prism.

Always work in two stages:

1. Find the area of the cross-section using the right 2D formula (rectangle, triangle, trapezium, etc.).
2. Multiply that area by the length of the prism.

For a triangular prism with a triangle of base $8\,\text{cm}$ and height $5\,\text{cm}$, lying $12\,\text{cm}$ long: area $= \tfrac{1}{2} \times 8 \times 5 = 20\,\text{cm}^2$, so volume $= 20 \times 12 = 240\,\text{cm}^3$.

A cylinder behaves exactly like a prism — the cross-section is just a circle. The area of a circle is $\pi r^2$, so: $V = \pi r^2 h,$ where $r$ is the radius of the circular end and $h$ is the height (the length of the cylinder).

Two practical tips:

• Read carefully: the question may give you the diameter, not the radius. Halve it before squaring.
• Decide on π early: leave the answer as a multiple of $\pi$ (e.g. $90\pi\,\text{cm}^3$), or round at the end using $\pi \approx 3.142$ or the $\pi$ button on your calculator.

Going the other way is also useful. If a cylinder of height $h$ has volume $V$, the radius is $r = \sqrt{V \div (\pi h)}$.

Surface area is the total area of every face on the outside of a 3D solid. It is measured in square units because each face is a 2D shape.

For a cuboid, there are six rectangles in three matching pairs. Two faces measure $l \times w$, two measure $l \times h$ and two measure $w \times h$, so: $\text{SA} = 2(lw + lh + wh).$

For a cube every face is identical, so $\text{SA} = 6a^2$.

For a cylinder, unroll it: the two ends are circles of area $\pi r^2$, and the curved side unrolls into a rectangle of width $h$ and length equal to the circle's circumference $2\pi r$. Adding them gives: $\text{SA} = 2\pi r^2 + 2\pi r h.$

The biggest skill is systematic counting: list every face, calculate each area, then add. If you forget one face, your final answer will be wrong by exactly that face's area.

A net is the flat shape you get if you unfold a 3D solid so that every face lies on the page. It is the most powerful way to find a surface area when no neat formula fits.

Standard nets to know:

• Cube: six identical squares (the classic cross of six squares is one valid net).
• Cuboid: six rectangles in three matching pairs.
• Triangular prism: two triangles and three rectangles.
• Cylinder: two circles connected by one rectangle (length $= 2\pi r$, height $= h$).

To find a surface area from a net, calculate the area of each piece, then add them all. This works for shapes with no tidy formula — you just need to recognise every face. Drawing the net is also a good way to check that opposite faces match.

Mastering volume and surface area for cuboids, prisms and cylinders sets you up for the rest of secondary geometry and many real-world problems:

• Cones, pyramids and spheres in IGCSE build on the volume = base × height pattern.
• Compound solids (a cylinder on top of a cuboid, for example) just add and subtract simple volumes.
• Density, mass and volume problems link volume to mass via $\rho = m / V$.
• Real-world applications include packaging, fuel tanks, water containers, paint coverage, and architecture.

The thinking is always the same: identify the cross-section, find its area, multiply by the length — or list every face, find its area, then add. Get those two routines fluent now and the rest of solid geometry follows easily.