| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | $lwh$ | $2(lw + lh + wh)$ |
| Cube | $a^3$ | $6a^2$ |
| Prism | (cross-section area) $\times$ length | perimeter $\times$ length $+$ 2 (end areas) |
| Cylinder | $\pi r^2 h$ | $2\pi r h + 2\pi r^2$ |
| Pyramid | $\tfrac{1}{3}\times$ base area $\times h$ | base $+$ areas of triangular faces |
| Cone | $\tfrac{1}{3}\pi r^2 h$ | $\pi r l + \pi r^2$ |
| Sphere | $\tfrac{4}{3}\pi r^3$ | $4\pi r^2$ |
| Hemisphere | $\tfrac{2}{3}\pi r^3$ | $3\pi r^2$ (curved $2\pi r^2$ + base $\pi r^2$) |
For a cone, slant height $l = \sqrt{r^2 + h^2}$.
Frustum of a cone (radii $R$ and $r$, height $h$)
$$V = \tfrac{1}{3}\pi h\,(R^2 + Rr + r^2)$$ Find it as (big cone) $-$ (small cone removed). Curved surface area $= \pi l (R + r)$ where $l$ is the slant height of the frustum.
Capacity and mass
- $1\ \text{cm}^3 = 1\ \text{ml}$, $\quad 1000\ \text{cm}^3 = 1\ \text{litre}$, $\quad 1\ \text{m}^3 = 1000\ \text{litres}$. $$\text{Density} = \frac{\text{mass}}{\text{volume}} \qquad \text{Mass} = \text{density}\times\text{volume}$$