1. Stepping into 3D
Up to now you've measured flat shapes — a rectangle's area, a circle's circumference. A solid is what you get when a flat shape gains a third dimension. Two new quantities come with that step.
The total area of every face of a solid. If you could unfold the solid into a flat net, the surface area is the area of that net. Measured in square units ($\text{cm}^2$, $\text{m}^2$, ...).
How much three-dimensional space a solid encloses — how many unit cubes you'd need to fill it. Measured in cubic units ($\text{cm}^3$, $\text{m}^3$, ...).
The units encode the dimensions. Surface area multiplies two lengths, so it ends in "squared." Volume multiplies three lengths, so it ends in "cubed." Mix them up and the answer is meaningless: nobody buys $300\,\text{cm}^2$ of milk.
Surface area is paint. Volume is water. Paint covers the outside; water fills the inside. Two completely different jobs, two completely different formulas, two completely different units.
2. Rectangular prisms (and cubes)
A rectangular prism — sometimes called a cuboid — is a box. Six rectangular faces, paired off into three matching pairs. Length $\ell$, width $w$, height $h$.
Volume is the easy one. Stack $h$ copies of a $\ell \times w$ rectangle on top of each other and you get
$$ V = \ell\,w\,h $$Surface area is the sum of three matching pairs of rectangles: top-and-bottom ($\ell w$ each), front-and-back ($\ell h$ each), left-and-right ($w h$ each). Doubling each gives
$$ S = 2\bigl(\ell w + \ell h + w h\bigr) $$A cube is the special case where every edge is the same length $s$. Substitute $\ell = w = h = s$ and the formulas collapse:
$$ V_{\text{cube}} = s^3 \qquad S_{\text{cube}} = 6 s^2 $$One way to see the cube's surface area without algebra: it has six identical square faces, each of area $s^2$. So $6 s^2$.
3. Cylinders
A cylinder has two circular ends (top and bottom) and one curved side wrapping between them. Radius $r$, height $h$.
Volume is the same trick as the cuboid: stack $h$ thin disks of area $\pi r^2$ on top of each other.
$$ V = \pi r^2 h $$For surface area, imagine peeling the cylinder open. The two end-caps are circles of area $\pi r^2$ each — together $2\pi r^2$. The curved side, when unrolled, is a rectangle: its height is $h$, and its width is the circumference of the base, $2\pi r$. So the lateral surface area is $2\pi r h$.
$$ S = 2\pi r^2 + 2\pi r h $$The first term is "top plus bottom"; the second is "the wrapper." A topless cylinder — say, a soup can with the lid removed — would have $\pi r^2 + 2\pi r h$. Always think about which faces are actually there.
4. Cones
A cone is what you get when you shrink a circular base down to a single point — the apex. Two quantities matter: the vertical height $h$ (straight from apex down to the centre of the base) and the slant height $\ell$ (along the curved surface from apex to the rim).
These are not the same. They're connected by the Pythagorean theorem, because together with the radius $r$ they form a right triangle inside the cone:
$$ \ell^2 = r^2 + h^2 $$Volume of a cone is one-third the volume of the cylinder that just contains it:
$$ V = \tfrac{1}{3}\,\pi r^2 h $$Surface area is the base (a circle, $\pi r^2$) plus the curved lateral surface, which unrolls into a sector of a circle and works out to $\pi r \ell$:
$$ S = \pi r^2 + \pi r \ell $$Volume uses the vertical height $h$ — measure straight down from the apex. Surface area uses the slant height $\ell$ — measure along the surface. Plugging $h$ into the surface-area formula is one of the most common mistakes on a geometry exam. If you only have $h$ and $r$, compute $\ell = \sqrt{r^2 + h^2}$ first.
5. Pyramids
A pyramid is a cone with corners — a flat polygonal base meeting at a single apex. The volume formula is just as forgiving as the cone's:
$$ V = \tfrac{1}{3}\,(\text{base area})\,h $$where $h$ is again the vertical height from the apex down to the plane of the base. For a square pyramid with side length $a$, base area is $a^2$, so $V = \tfrac{1}{3} a^2 h$.
Three identical square pyramids, glued together correctly, fill exactly one cube of the same base and height. So one pyramid is $\tfrac{1}{3}$ of the cube — and by an extension of that argument (running through general "cone-shaped" solids), every pyramid and cone is one-third the prism or cylinder that contains it. The factor isn't a coincidence; it's geometry.
Surface area depends on the shape of the base. For any pyramid it's the base area plus the area of every triangular lateral face. For a square pyramid with base side $a$ and slant height $\ell_s$ (apex-to-midpoint-of-base-edge, not apex-to-corner), the four triangular faces each have area $\tfrac{1}{2} a \ell_s$, so
$$ S_{\text{square pyramid}} = a^2 + 2 a \ell_s $$There's no single tidy formula for "every pyramid" — for a triangular base, a pentagonal base, or anything irregular, you sum the base area and each triangular face explicitly.
6. Spheres
A sphere is the cleanest 3D shape: every point on its surface is exactly $r$ from the centre. Just one parameter — the radius. And just two formulas.
$$ S = 4\pi r^2 \qquad V = \tfrac{4}{3}\,\pi r^3 $$The surface area can be remembered as "four great-circle areas." A great circle through the centre of the sphere has area $\pi r^2$, and the sphere's total skin is exactly four of those. The volume is $\tfrac{4}{3}\pi r^3$, with the cube of $r$ giving away its three-dimensional nature.
Archimedes of Syracuse (c. 287–212 BCE) discovered that a sphere has exactly two-thirds the volume — and two-thirds the surface area — of the smallest cylinder that contains it. He was so proud of this result that he asked for a diagram of a sphere inscribed in a cylinder to be carved on his tombstone. Roughly 137 years later, Cicero, as quaestor in Sicily, tracked down the neglected grave and confirmed the carving was still there. The result is the foundation of every formula in this section.
7. Scaling, composites, and units
Two short ideas that show up everywhere — and one practical reminder about the units you'll actually meet in the wild.
The square-cube law
Scale every length of a solid by a factor $k$. Areas — and surface area is just a sum of areas — scale by $k^2$. Volumes scale by $k^3$.
$$ L \to kL \quad\Longrightarrow\quad S \to k^2 S \quad\Longrightarrow\quad V \to k^3 V $$Double the radius of a sphere and its surface area quadruples while its volume goes up eight-fold. This isn't an arithmetic curiosity: it's why ants can fall any distance and walk away, why elephants need columnar legs, and why crushed ice cools a drink faster than a single block of the same mass.
Composite solids
Most real objects aren't pure cylinders or pure cones — they're combinations. A grain silo is a cylinder topped by a hemisphere. A pencil is a cylinder with a cone on the end. The recipe is mechanical:
- Volume. Add the volumes of the parts. (Subtract for hollowed-out pieces.)
- Surface area. Sum the exposed faces only — drop any surface that's hidden where two parts meet.
The silo's surface area is the cylinder's lateral surface ($2\pi r h$) plus its flat bottom ($\pi r^2$) plus the hemisphere's curved surface ($2\pi r^2$) — but not the cylinder's top, because the hemisphere is sitting on it.
Real-world units
Outside of textbook problems, volumes show up in litres and gallons as often as in cubic centimetres. The conversions worth knowing:
$$ 1\,\text{L} = 1000\,\text{cm}^3 \qquad 1\,\text{m}^3 = 1000\,\text{L} \qquad 1\,\text{US gallon} \approx 3.785\,\text{L} $$So a $2$-litre bottle holds $2000\,\text{cm}^3$, and a cubic-metre tank holds about $264$ US gallons. Always convert before plugging into a formula — mixing $\text{cm}$ with $\text{m}$ inside a single calculation is the silent error that ruins answers.