• Volume = amount of 3D space a shape fills. Measured in cubic units (cm³, m³...).
• Cuboid volume: V = lwh — the only volume formula you must memorise; everything else is given in the exam.
• Any prism (constant cross-section): V = A × l (cross-sectional area × length).
• Cylinder is really just a prism with a circular cross-section: V = πr²h.
• Pyramid and cone volumes are both "⅓ × base area × height" — they shrink to a point, so you divide by 3.
• Sphere volume: V = (4/3)πr³.
• Real exam shapes are rarely "pure" — they're often compound (add volumes), a fraction of a shape (e.g. hemisphere = half sphere), or a frustum (big cone minus small cone).
• Surface area = sum of the areas of every face. For flat-faced solids, sketch the net and add up each face's area.
• Cylinder curved surface: A = 2πrh. Cone curved surface: A = πrl (l = slant height, not perpendicular height!). Sphere: A = 4πr².

A cuboid is just a fancy name for a rectangular box — think of a shoebox or a brick. A cube is a special cuboid where all three edges (length, width, height) are exactly equal, like a dice.

To find how much space is inside, you're really asking: "how many 1cm × 1cm × 1cm cubes could I pack inside this box?" Picture building up layers — one flat layer covers the base (length × width), and then you stack up h of those layers. That's exactly why the formula multiplies all three dimensions together.

A prism is any 3D shape that has the exact same 2D cross-section running all the way through it, like a loaf of bread — slice it anywhere along its length and you get an identical shape. A cuboid is actually just a prism with a rectangular cross-section!

This is the single most useful idea in the whole chapter: volume of a prism = area of the cross-section × the length of the prism. It doesn't matter what shape the cross-section is — a triangle, an L-shape, a trapezium, even a weird compound shape — as long as you can find its area, you can find the volume.

If you're given the volume and length, you can rearrange to find the cross-sectional area: A = V ÷ l.

A cylinder (think of a soup can) is just a prism whose cross-section happens to be a circle. Since the area of a circle is πr², and volume of a prism is area × length, you just swap "length" for "height" and you get the cylinder formula.

Both pyramids and cones taper to a single point (an apex) instead of continuing at a constant cross-section like a prism. Imagine three identical pyramids fitting together to exactly fill a cuboid of the same base and height — that's genuinely where the ⅓ comes from. So the pattern is simple: take the "prism version" of the shape's volume formula, and multiply by ⅓.

A cone is just a pyramid with a circular base, so its formula is the pyramid formula with A = πr² substituted in.

A sphere (a perfectly round ball) has just one formula to know for volume. Unlike the shapes above, it doesn't come from a simple "base × height" idea — but you're always given it in the exam, so you just need to be confident substituting into it.

In the exam, shapes are almost never a "pure" cuboid or cone sitting on their own. Instead, they tend to fall into one of three disguises. Spotting which disguise you're dealing with is honestly half the battle — once you know which category a question falls into, the maths itself is usually just the formulas from Topic 1.

If the 3D shape has an L-shaped, cross-shaped, or otherwise "compound" 2D cross-section running through it, treat it exactly like a prism: find the area of that (possibly weird) cross-section by splitting it into rectangles/triangles, then multiply by the length.

Sometimes you only get part of a shape — most commonly a hemisphere (exactly half a sphere — think of a bowl). Just calculate the volume of the "full" version of the shape, then take the fraction you need.

A trickier version of this is a frustum — a cone or pyramid with its pointed top sliced off (imagine a lampshade, or a bucket). To find its volume, you find the volume of the full, un-sliced cone/pyramid, then subtract the volume of the small cone/pyramid that was removed from the top.

This is when two (or more) standard 3D shapes are stuck together — like an ice cream cone with a hemisphere of ice cream on top, or a pencil (cylinder + cone). Just find the volume of each separate solid using the formulas from Topic 1, then add them together.

Surface area is simply the total area you'd need to wrap around the outside of a shape — it's a 2D idea (area) being applied to every face of a 3D object. For any solid made of flat faces, the strategy is always the same:

• Imagine (or sketch) the net — the shape "unfolded" flat.
• Work out the area of each individual face.
• Add every face's area together.

For example, a square-based pyramid (with the apex directly above the centre of the base) unfolds into a net of one square base plus four identical isosceles triangles. You'd calculate the area of the square, calculate the area of one triangle, then add the square to four lots of the triangle.

A cylinder has two flat circular faces (top and bottom) and one curved surface wrapped around the middle. If you unroll that curved surface, it flattens out into a rectangle — its height is the same as the cylinder's height, and its width is exactly the circumference of the circular base (2πr), because that's the distance it has to wrap around.

A cone has one flat circular base and one curved surface. If you unroll the curved surface, it becomes a sector (a "pizza slice" shape) of radius equal to the slant height, l — not the perpendicular height. This is the same trap as with volume, but even more important here, because the curved surface area formula only works with the slant height.

A sphere has just one continuous curved surface — no flat faces at all, since it's perfectly round all the way around.

A hemisphere is exactly half of a sphere sliced through the middle. That means it has half the curved surface area of a full sphere — plus a brand new flat circular face where it was sliced (which a full sphere doesn't have at all). Don't forget this flat circle — it's the single most commonly forgotten piece in the whole surface area topic!

Everything tagged Given is printed on your formula sheet — don't waste energy memorising it, just practice using it. Everything tagged Not given you genuinely need in your head.