⚽ Sphere, Pyramid & Solids

• Sphere: volume $= \dfrac{4}{3}\pi r^3$ and surface area $= 4\pi r^2$.

• Hemisphere (half a sphere): volume $= \dfrac{2}{3}\pi r^3$, curved surface $= 2\pi r^2$, and total surface $= 3\pi r^2$ (the curved part plus the flat circular lid). Read solid vs hollow carefully — a bowl has no lid, so it uses only the curved $2\pi r^2$.

• Pyramid (pointed top on a flat base): volume $= \dfrac{1}{3}\times\text{base area}\times h$, and surface $= \text{base} + \dfrac{1}{2}\times\text{perimeter}\times l$, where $l$ is the slant height. Always use the slant height for the faces, not the vertical height.

• Prism (same cross-section all the way up): volume $= \text{base area}\times h$. There is no one-third here — that is the single biggest trap, because a pyramid is exactly one-third of the prism on the same base.

• Combination solids: a shape built from parts (an ice-cream cone is a cone topped by a hemisphere; a capsule is a cylinder with a hemisphere at each end). Just add the volumes of the parts, and add only the exposed surfaces.

• Melting & recasting: when a solid is melted and reshaped, the volume stays the same. So set $\text{volume before} = \text{volume after}$ and solve. For $n$ identical small pieces, $\text{big volume} = n\times\text{small volume}$; the $\pi$ (and any other constant) cancels straight away.

• Ratios of similar solids: volume grows as the cube of length, surface as the square. So radii in ratio $a:b$ give volumes in ratio $a^3:b^3$ and surface areas in ratio $a^2:b^2$.