Round the base off and nothing else changes. A cylinder is a prism and a cone is a pyramid — the formulas follow from that sentence.
How much does it hold?
A cylindrical drum has radius 20 cm and height 50 cm. Take π ≈ 3.14.
V = πr²h = 3.14(400)(50) = 62 800 cm³, which is 62.8 litres. B uses 2πrh, the curved surface area, not the volume — the two differ by a factor of r2 and are easy to grab the wrong way. D doubles r into the formula, using the diameter 40 in place of the radius.
V = πr²h
This is still V = Bh. The base area B is πr² because the base is a circle, and nothing else about the method changes: identical circular layers, stacked h deep.
Unroll the curved surface and it is a rectangle: its height is h, and its width is once round the circle, the circumference 2πr. So LSA = 2πrh, and capping both ends gives TSA = 2πrh + 2πr².
LSA = 2πrh = 2(227)(7)(12)
the 7s cancel
LSA = 528 cm²
the curved side
2B = 2(227)(49) = 308 cm²
the two flat ends
TSA = 528 + 308 = 836 cm²
a tin can, in full
Use 227 whenever the radius is a multiple of 7, and 3.14 otherwise. Choosing to match the numbers keeps the arithmetic exact and the answer clean.
V = ⅓πr²h LSA = πrℓ TSA = πrℓ + πr²
The same one third as the pyramid, for the same reason: three cones fill the cylinder on the same base and height. And the same two heights are in play — volume uses the vertical h, curved surface uses the slant ℓ.
For a cone the right triangle is simpler than a pyramid’s: no halving. The radius already runs from the centre to the edge, so ℓ² = h² + r² outright.
V = ⅓π(36)(8) = 96π cm³
≈ 301.6 cm³
ℓ = √8² + 6² = √100 = 10
the 6-8-10 triangle
LSA = π(6)(10) = 60π cm²
the curved surface
TSA = 60π + 36π = 96π cm²
≈ 301.6 cm²
The volume and the surface area come out as the same multiple of π here, which is a coincidence of these numbers — and a reminder to keep the units attached. One is cm³ and the other cm²; they are not the same quantity.
| Volume | Lateral area | Total area | |
|---|---|---|---|
| Cylinder | πr²h | 2πrh | 2πrh + 2πr² |
| Cone | ⅓πr²h | πrℓ | πrℓ + πr² |
Count the lengths. πrℓ multiplies two lengths, so it is square units. πr²h multiplies three, so it is cubic. If a question asks for litres and your expression has only two lengths in it, you have reached for the wrong formula — and this check works on every formula in the unit.
Substitute the volume and every length you know, divide both sides by π, and solve what is left. If the unknown is the radius, finish with a square root.
A cylindrical tin holds 245π cm³ and is 5 cm tall. Find its radius.
245π = πr²(5)
V = πr²h
r² = 245 ÷ 5 = 49
π cancels from both sides
r = 7 cm
check: π(49)(5) = 245π ✓
A cone has volume 32π cm³ and radius 4 cm. Find its height.
32π = ⅓π(16)h
substitute first
96 = 16h
cancel π, times 3
h = 6 cm
check: ⅓π(16)(6) = 32π ✓
Stopping one line early gives “r = 49 cm”. Put it back and the tin would hold π(2 401)(5) = 12 005π cm³, nowhere near 245π. When the radius is the unknown, the last move is always a square root.
The same four formulas, with one piece added, taken away or found first.
A hollow pipe: outer radius 5 cm, inner radius 3 cm, 20 cm long. Volume of material?
B = π(5²) − π(3²) = 16π cm²
a ring: big circle minus the hole
V = 16π × 20 = 320π cm³
still V = B h
Subtract the areas, not the radii: π(5 − 3)² = 4π is far too small.
A cone with radius 5 cm and slant height 13 cm. Find its volume.
h = √13² − 5² = √144 = 12
the slant is the hypotenuse
V = ⅓π(25)(12) = 100π cm³
volume needs the vertical h
Here the slant is given, so subtract to reach the height.
A pail with no lid: radius 10 cm, height 30 cm, π ≈ 3.14. How much sheet metal?
LSA = 2(3.14)(10)(30) = 1 884 cm²
the curved side
one base = 3.14(100) = 314 cm²
a bottom only
total = 2 198 cm²
2πrh + πr², not + 2πr²
An open container loses one circle; a pipe open at both ends loses both.
Expect a tank, can or ice-cream cone with two measurements given and a volume, capacity or curved area asked, often “in terms of π” or with π ≈ 3.14. Find any missing length with ℓ² = h² + r² first, then use the formula that needs it. The 10-second check: count the lengths in your formula (two for area, three for volume), and a cone must come out as one third of its cylinder.
Find the curved surface area
A cone has radius 9 cm and height 12 cm. Leave your answer in terms of π.
ℓ = √12² + 9² = √225 = 15 cm, so LSA = πrℓ = π(9)(15) = 135π cm². A uses the height in place of the slant, π(9)(12) — and since the slant must exceed the height, 108π is too small. C is the volume divided by π-free thinking, ⅓π(81)(12) = 324π, which is cm³. D adds the base circle to A.
Hint: a cone’s slant height needs no halving — ℓ² = h² + r² straight off. And “curved surface” means the base circle is not included.
