Five solids, two ideas: base × height, and a third of it.
Find the volume
A cylinder has radius 3 cm and height 10 cm.
V = πr²h = π(3²)(10) = 90π cm³. D is 2πrh, the curved surface area — an area formula wearing a volume's clothes. Volume always squares the radius.
A prism or cylinder is the same slice stacked all the way up.
V = lwh
V = πr²h
V = B × h
base = π(3)² = 9π
the circular end
V = 9π × 10 = 90π cm³
≈ 283 cm³
The height is the perpendicular distance between the two ends, not the slanted edge. That distinction matters for pyramids and cones especially.
V = ⅓ × B × h
V = ⅓πr²h
A cone of the same base and height holds exactly one third of the cylinder — it takes three coneloads to fill it.
Dropping the ⅓ triples the answer, and multiple-choice questions almost always include that value as one of the options.
Equal volumes, equal heights. A cone and a cylinder of the same height hold the same amount only if the cone’s base is 3 times as big: ⅓πR²h = πr²h gives R² = 3r². With a cylinder of radius 5, R² = 75 and R = √75 = 5√3 — take the square root of the 3 as well.
Unroll the cylinder and it becomes two circles and a rectangle.
two ends = 2π(3)² = 18π
the circles
side = 2π(3)(10) = 60π
the rectangle
total = 78π cm²
≈ 245 cm²
| Solid | Volume | Surface area |
|---|---|---|
| cube | s³ | 6s² |
| rectangular prism | lwh | 2(lw + lh + wh) |
| cylinder | πr²h | 2πr² + 2πrh |
| cone | ⅓πr²h | πr² + πrl |
| pyramid | ⅓Bh | B + ½(perimeter × slant) |
| sphere | ⁴⁄₃πr³ | 4πr² |
| any prism (base B) | B × h | 2B + (perimeter of base) × h |
A cone's volume uses the vertical height h; its curved surface uses the slant height l. They are two legs and the hypotenuse of a right triangle: l² = r² + h².
Double every length and the other measures do not double.
length
× 2
area
× 4
volume
× 8
Area involves two dimensions, so the factor is squared. Volume involves three, so it is cubed.
A common exam question: “the radius is tripled, what happens to the volume?” The answer is 27 times, not 3.
When the dimensions change by different amounts, multiply each factor into the formula. Cylinder with radius × 3 and height ÷ 3: V = π(3r)²(h3) = 9 × ⅓ × πr²h, so the volume becomes 3 times as large.
A solid ball of radius 6 cm sinks completely in a cylindrical tank of radius 8 cm. How far does the water rise? The rise is a thin cylinder with the same volume as the ball.
ball = ⁴⁄₃π(6)³ = 288π
6³ = 216, × 4 ÷ 3
rise = π(8)² × h = 64πh
a cylinder of the tank’s radius
64πh = 288π
volumes equal
h = 4.5 cm
288 ÷ 64
The rise uses the tank’s base area, πR². Writing 8πh drops the square and gives 36 cm, which is taller than the ball itself — impossible.
Combined solids just add (or subtract) their parts. A cylinder of radius 3 and height 5 with a cone of the same radius and height 4 on top: 45π + ⅓ × 9π × 4 = 45π + 12π = 57π.
An open-top cylindrical can has radius 5 cm and height 8 cm. Find the area of metal used.
bottom only = π(5)² = 25π
one circle, not two
side = 2π(5)(8) = 80π
the rectangle
total = 105π cm²
≈ 330 cm²
Different: the formula 2πr² + 2πrh assumes both lids. Build the net from the faces that actually exist.
A solid hemisphere has radius 4 cm. Find its total surface area.
curved = ½ × 4π(4)² = 32π
half a sphere’s surface
flat face = π(4)² = 16π
the circle where it was cut
total = 48π cm²
3πr²
Halving 4πr² gives 32π, which leaves out the flat disc. A solid hemisphere has a curved part and a flat part: 2πr² + πr² = 3πr².
A tank is 40 cm long, 25 cm wide and 30 cm deep. How many litres does it hold?
V = 40 × 25 × 30 = 30 000 cm³
lwh
1 L = 1000 cm³
a 10 cm cube
30 000 ÷ 1000 = 30 L
and 1 m³ = 1000 L
Different: the answer is asked in litres, not cm³. Work out the volume in one unit, then convert once at the end.
Many items give one measure and ask for another. Find the length first, then go forward.
| Given | Find the length | Then |
|---|---|---|
| cube, surface 96 | 6s² = 96 → s = 4 | V = 4³ = 64 |
| cone, V = 96π, h = 8 | ⅓πr²(8) = 96π → r² = 36, r = 6 | slant = √36 + 64 = 10 |
| cone, r = 6, slant 10 | h = √100 − 36 = 8 | V = ⅓π(36)(8) = 96π |
Expect the twist, not the formula: a slant height where the volume needs the vertical height, a surface area that hides the side, a ball dropped in water, a solid built from two shapes, or a scaling question. Find the missing length first (often with Pythagoras), then apply the formula. Ten-second check: volume answers are in cubic units, and a cone is a third of its cylinder.
Find the volume
A cone has radius 6 cm and height 8 cm.
V = ⅓πr²h = ⅓π(36)(8) = 96π cm³. A forgets the ⅓ and gives the cylinder that would enclose the cone — the cone holds exactly a third of it.
