🥫 Cylinder, Cone & Frustum

• Volume $= \pi r^2 h$ — the base area $\pi r^2$ times the height. Find the base area first, then multiply by $h$.

• Curved surface area (CSA) $= 2\pi r h$ — only the side, with no top or bottom lid.

• Total surface area (TSA) $= 2\pi r h + 2\pi r^2 = 2\pi r(r+h)$ — the side plus both circular ends. Always use the factored form $2\pi r(r+h)$; it is one multiplication and stops you dropping the two lids.

• Slant height $l = \sqrt{r^2 + h^2}$ — the slant is the hypotenuse of the right triangle made by the radius and the vertical height. Lean on the Pythagoras triplets $(3,4,5),(5,12,13),(8,15,17),(7,24,25)$.

• Volume $= \dfrac{1}{3}\pi r^2 h$ — exactly one-third of the cylinder that shares the same base and height (it takes 3 cones of water to fill that cylinder). Never forget the $\dfrac{1}{3}$.

• CSA $= \pi r l$, and TSA $= \pi r l + \pi r^2 = \pi r(l + r)$ — curved face plus the circular base. Use the slant height $l$ here, never the vertical height.

• Slant height $l = \sqrt{h^2 + (R-r)^2}$ — note it uses the difference of the radii, not the sum.

• Volume $= \dfrac{1}{3}\pi h(R^2 + Rr + r^2)$ — keep the middle $Rr$ term; it is the most common thing to drop.

• CSA $= \pi l(R + r)$ (the radii enter as the SUM here), and TSA $= \pi l(R+r) + \pi R^2 + \pi r^2$ — the curved side plus the big and small circular ends.

The exam habit to build: take $\pi=\dfrac{22}{7}$ whenever a radius is a multiple of 7 and cancel that 7 before multiplying — for example $\dfrac{22}{7}\times 7^2 = 22\times 7 = 154$. Cancel first, multiply last.