These three solids are all prisms, so one rule covers all of them: volume = area of cross-section × length. For a cuboid that gives lwh, for a cylinder it gives πr²h. Surface area follows the same logic: the two ends plus the wrapper, so a cylinder is 2πr² + 2πrh. None of these are given on the O-Level formula sheet, unlike the cone and sphere, so they have to be recalled.
The formula sheet gives you the cone and the sphere. It does not give you the cuboid, the prism or the cylinder, on the assumption that those are the easy ones. Students spend their revision on the two formulas they will be handed and lose marks on the three they will not.
The good news is that you do not need to memorise six separate formulas. All three solids are prisms, and one rule generates every one of them.
The rule
volume = cross-section area × length
Cuboid
V = lwh, SA = 2(lw + lh + wh)
Cylinder
V = πr²h
Cylinder curved SA
2πrh
Cylinder total SA
2πrh + 2πr²
Any prism SA
2 × cross-section + perimeter × length
A prism is any solid with the same cross-section all the way through. A cuboid is a prism with a rectangular cross-section. A cylinder is a prism with a circular one. Once you see that, the volume formula is the same sentence every time and only the shape of the end face changes.
A prism has a right-angled triangular cross-section with base 6 cm and height 8 cm. The prism is 15 cm long. Find its volume and total surface area. The hypotenuse of the triangle is 10 cm.
Solution
½ × 6 × 8 = 24 cm².= 24 × 15 = 360 cm³.2 × 24 = 48 cm².6 + 8 + 10 = 24 cm.= perimeter × length = 24 × 15 = 360 cm².= 48 + 360 = 408 cm².Why the wrapper rule works: imagine unrolling the sides of the prism flat. You get a rectangle whose height is the length of the prism and whose width is the perimeter of the cross-section. That is why it is always perimeter multiplied by length, for every prism including the cylinder.
A closed cylindrical tin has radius 7 cm and height 12 cm. Find its volume and its total surface area, taking π = 22/7.
Solution
= πr²h = 22/7 × 7² × 12.= 22/7 × 49 × 12 = 22 × 7 × 12 = 1848 cm³.= 2πrh = 2 × 22/7 × 7 × 12 = 528 cm².2πr² = 2 × 22/7 × 49 = 308 cm².= 528 + 308 = 836 cm².The step students get wrong
Adding the ends when the question did not ask for them. An open pipe has no ends, an open-topped tank has one, and a closed tin has two. Read the wording: "curved surface area" means the wrapper only, and "open" always means you leave a face out. Adding 2πr² by reflex is one of the most common lost marks in mensuration.
A cuboid has a square base of side x cm and a height of 10 cm. Its volume is 640 cm³. Find x.
Solution
= lwh, and the base is square so l = w = x.x × x × 10 = 640, so 10x² = 640.x² = 64.x = 8 cm. Reject x = -8 because a length cannot be negative.That last line matters. Any question where you square-root to find a length has two algebraic answers and one physical one, the same reasoning you apply when solving a quadratic equation in a real-world context.
Which of these formulas are on the formula sheet?
None of them. The O-Level sheet gives you the curved surface area and volume of a cone, and the surface area and volume of a sphere. Cuboids, prisms and cylinders are assumed knowledge, so they must be recalled from memory.
What is the difference between curved and total surface area?
Curved surface area is the wrapper only, 2πrh for a cylinder. Total surface area adds the flat ends. Whether you add none, one or two of those ends depends entirely on whether the solid is open or closed, which the question will state.
Is a cylinder really a prism?
For the purposes of these formulas, yes. It has the same cross-section all the way through, so volume = cross-section area × height applies, giving πr²h. Strictly, a prism has flat polygonal faces, but the volume rule works identically.
How do I handle a solid made of two shapes joined together?
Split it into parts, work out each part separately, then add the volumes. Surface area needs more care because the joining faces disappear from the outside. That is covered in composite solids.
How does this connect to the rest of E-Maths?
Cross-section area is just perimeter and area applied to the end face, so weak 2D area work shows up immediately here. If a question scales a solid up or down, the volume changes by the cube of the scale factor, covered in area and volume scale factor.
— Mr Gan Math Tuition
Mr. Gan teaches the cross-section rule so you can rebuild these formulas instead of memorising six of them.
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