Summary — everything in this chapter, at a glance
- 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.
Topic 1 — Volume of Standard Solids
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.
Watch the wording
You'll sometimes see "depth" used instead of "height", or "breadth" used instead of "width".
Don't let the vocabulary throw you — it's still just three perpendicular edges multiplied together.
Practice Question
A cuboid-shaped water tank has length 1.2 m, width 80 cm, and height 60 cm. Find its volume in cm³.
Prisms — the master idea behind everything else
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.