Cylinder, cone & sphere

For a cuboid, volume is easy: length times width times height. As soon as something is round, formulas often get guessed. That is not necessary, because cylinder, cone and sphere belong together. All three formulas contain the circle area $\pi r^2$, and the differences fit into a single sentence.

The cylinder: circle times height

A cylinder is a can: a circle at the bottom, the same circle on top, straight walls in between. Exactly as for a cuboid it is base area times height, except the base is now a circle. So $V = \pi r^2 \cdot h$. A can with $r = 4$ cm and $h = 10$ cm has a volume of $\pi \cdot 16 \cdot 10 \approx 503$ cm³, so about half a litre.

For the surface of a can you need three parts: two circles and the wall. Unrolled, the wall is a rectangle. Its height is $h$, and its width is the circumference $2 \pi r$, because that is exactly how far it reaches around. So $O = 2 \pi r^2 + 2 \pi r h$.

The cone: a third of it

A cone has the same base as a cylinder but tapers to a point. It fits exactly three times into the cylinder of the same height. That is why the formula has a third in it: $V = \frac{1}{3} \pi r^2 h$. You can genuinely test this by filling a cone-shaped cup with water three times and pouring it into a cylindrical one of the same height and radius.

A cone has two different lengths, and mixing them up is the classic mistake. The height $h$ runs vertically from the middle of the circle to the tip. The slant height $s$ runs from the edge of the circle to the tip and is always longer. The two are linked by Pythagoras: $s^2 = r^2 + h^2$. For the volume you need $h$, for the surface you need $s$.

The sphere: only the radius counts

A sphere has no height and no base, it only has a radius. Accordingly both formulas contain nothing but $r$ and $\pi$: $V = \frac{4}{3} \pi r^3$ and $O = 4 \pi r^2$. The surface formula is easier to remember than it looks: it is exactly four times the circle area. So a football with $r = 11$ cm has a surface of $4 \pi \cdot 121 \approx 1521$ cm².