Surface area and volume — cones and spheres

In this subtopic, we'll learn how to calculate the surface area and volume of two important 3D shapes: cones and spheres. You encounter these shapes every day — ice cream cones, traffic cones, basketballs, oranges, and even planet Earth!

Before diving in, let's recall two important ideas:

• Surface area is the total area covering the outside of a 3D shape (imagine painting every part of its surface — how much paint would you need?).
• Volume is the amount of space inside a 3D shape (imagine filling it with water — how much water would it hold?).

Both measurements require specific formulas, and you'll need to be comfortable working with $\pi$ (pi ≈ 3.14159...).

A cone is a 3D shape with a circular base that tapers to a single point called the apex (or vertex).

A cone has three key measurements:

• Radius ($r$) — the radius of the circular base
• Height ($h$) — the perpendicular (vertical) distance from the centre of the base to the apex
• Slant height ($l$) — the distance from the edge of the base along the curved surface to the apex

These three measurements are related by the Pythagorean theorem:
$l=\sqrt{r^2+h^2}$

This works because the radius, height, and slant height form a right triangle!