What is a combination of solids? A combination of solids is a three-dimensional object formed by joining two or more basic solids. Very few real objects are pure shapes. An ice-cream cone is a cone topped by a hemisphere; a medicine capsule is a cylinder with a hemisphere on each end; a circus tent is often a cylinder capped by a cone; a childs spinning top is a cone fixed to a hemisphere. To handle such objects we break them into the standard solids we already know and recombine the relevant pieces.
The two golden rules. Surface area and volume behave very differently when solids are joined, and confusing them is the single most common board-exam mistake.
- Volume adds, always. Space inside two joined solids is simply the sum of the two volumes. Nothing is lost at the joint, because the joint is just a shared internal boundary: $V_{\text{total}}=V_1+V_2$.
- Surface area counts only what is exposed. When two solids are stuck together the faces that meet are sealed inside and vanish from view. So you add only the outer visible surfaces, never the full TSAs of both parts.
Standard formulas you must know cold.
| Solid | CSA / LSA | TSA | Volume |
|---|---|---|---|
| Cube (side $a$) | $4a^2$ | $6a^2$ | $a^3$ |
| Cuboid ($l,b,h$) | $2h(l+b)$ | $2(lb+bh+hl)$ | $lbh$ |
| Cylinder ($r,h$) | $2\pi rh$ | $2\pi r(r+h)$ | $\pi r^2 h$ |
| Cone ($r,h,l$) | $\pi r l$ | $\pi r(r+l)$ | $\tfrac13\pi r^2 h$ |
| Sphere ($r$) | $4\pi r^2$ | $4\pi r^2$ | $\tfrac43\pi r^3$ |
| Hemisphere ($r$) | $2\pi r^2$ | $3\pi r^2$ | $\tfrac23\pi r^3$ |
The cone always needs its slant height $l=\sqrt{r^2+h^2}$ for any surface-area work, while volume needs the vertical height $h$. Keep the two straight: never feed a slant height into a volume formula.
How joints change the surface area — worked reasoning.
- Cone on a hemisphere (a toy). The flat base of the cone sits exactly on the flat face of the hemisphere; both circular faces are hidden. Exposed surface $=$ CSA of cone $+$ CSA of hemisphere $=\pi r l + 2\pi r^2$.
- Cylinder with a hemisphere on each end (a capsule). Two flat ends of the cylinder are covered. Exposed surface $=$ CSA of cylinder $+ 2\times$ CSA of hemisphere $=2\pi r h + 2(2\pi r^2)=2\pi r h + 4\pi r^2$.
- Cylinder topped by a cone (a tent / a rocket). The top face of the cylinder is sealed by the cone base and the cone base is itself hidden. Exposed surface $=$ CSA of cylinder $+$ CSA of cone $+$ the bottom circle if the tent has a floor (usually it does not). For canvas of a tent: $2\pi r h + \pi r l$.
- Hemisphere scooped out of a cube / cuboid. A hemispherical cavity removes the circle $\pi r^2$ from one face but adds the inner curved surface $2\pi r^2$. Net surface $=6a^2-\pi r^2+2\pi r^2=6a^2+\pi r^2$.
- Two cubes joined. Joining two cubes of side $a$ into a cuboid hides one face of each cube ($2a^2$ in total): new TSA $=2(6a^2)-2a^2=10a^2$, not $12a^2$.
Capacity vs. volume. For containers (tanks, vessels, bottles) the volume in cubic centimetres can be converted to capacity: $1000\text{ cm}^3 = 1$ litre, and $1\text{ m}^3 = 1000$ litres. Always state the final answer in the unit the question asks for.
Common mistakes to avoid. (i) Adding TSAs instead of only the exposed surfaces. (ii) Forgetting that a hemisphere on a flat solid adds $2\pi r^2$ while only the circle $\pi r^2$ is lost. (iii) Mixing slant height $l$ into a volume. (iv) Using diameter where the formula wants radius — halve it first. (v) Forgetting the second hemisphere in a capsule.
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