🧊 Mixed 2D & 3D

Cut the figure into shapes you already know and add (or subtract) their areas. The three you need most:

• Triangle: $\dfrac{1}{2}\times b\times h$  ·  Rectangle: $l\times b$  ·  Circle: $\pi r^2$.

Angles still obey the basics — a triangle's angles sum to $180^\circ$, and corresponding parts of congruent or similar triangles match. The moment a right angle shows up (a diagonal, a perpendicular from a centre, a half-diagonal), use Pythagoras $a^2+b^2=c^2$.

• Cube (edge $a$): volume $a^3$, surface area $6a^2$, space diagonal $a\sqrt3$.

• Cuboid $l\times b\times h$: volume $lbh$, space diagonal $\sqrt{l^2+b^2+h^2}$.

• Cylinder: volume $\pi r^2 h$, curved surface $2\pi r h$.

• Cone: volume $\dfrac{1}{3}\pi r^2 h$, curved surface $\pi r l$ with slant $l=\sqrt{r^2+h^2}$.

• Sphere: volume $\dfrac{4}{3}\pi r^3$, surface $4\pi r^2$.  ·  Any prism: volume $=$ base area $\times$ length.

The longest rod in a box is its space diagonal — the same Pythagoras idea, now in three directions: $\sqrt{l^2+b^2+h^2}$.