Sphere basics

A sphere is a perfectly round three-dimensional shape, like a ball.
Every point on its surface sits the same distance from the center.
That distance is the radius, written $r$.

The diameter is the full width across the sphere through the center.
It is always twice the radius: $\text{diameter} = 2r$.
So if you are told the diameter, halve it to get the radius: $r = \frac{\text{diameter}}{2}$.

The volume (space inside) is $V = \frac{4}{3}\pi r^3$.
For a sphere with radius $3$, the volume is $\frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi$.
Notice the radius is cubed, so always work out $r^3$ first.

The surface area (the outside skin) is $SA = 4\pi r^2$.
Here the radius is squared, not cubed.
Keeping these two formulas straight is the main thing this topic tests.

One common setup is a sphere that fits snugly inside a cube.
The sphere touches each face of the cube, so its diameter equals the side length of the cube.
That lets you find the radius as half the cube's side.

Find the volume of a sphere with radius $2$.
Use $V = \frac{4}{3}\pi r^3$ with $r = 2$.
So $V = \frac{4}{3}\pi (2)^3 = \frac{4}{3}\pi (8) = \frac{32}{3}\pi$.

Find the surface area of a sphere with radius $3$.
Use $SA = 4\pi r^2$ with $r = 3$.
So $SA = 4\pi (3)^2 = 4\pi (9) = 36\pi$.

A sphere is inscribed inside a cube with side length $10$. What is the sphere's radius?
An inscribed sphere touches every face, so its diameter equals the cube's side, $10$.
The radius is half the diameter: $r = \frac{10}{2} = 5$.