Find the volume of a cube, sphere, cylinder or cone.
Volume is the space a three-dimensional object occupies, in cubic units. A cuboid is length times width times height; a cylinder is pi times radius squared times height; a sphere is four-thirds pi times radius cubed; a cone is one-third the volume of the cylinder that contains it. That last relationship is worth remembering — a cone holds exactly a third of the cylinder with the same base and height, which is why three conical measures fill one cylindrical one.
Scaling is where intuition fails hardest. Volume grows with the cube of a linear dimension, so doubling the radius of a tank multiplies its capacity by eight, not two. A pizza twice the diameter has four times the area but a fixed thickness, while a ball twice the diameter holds eight times as much. This is also why converting units catches people out: one cubic metre is a million cubic centimetres, and one litre is a thousand cubic centimetres, so a small error in a linear measurement becomes a large error in the volume it produces.
One thousand. A litre is defined as 1,000 cubic centimetres, and a cubic metre is one million cubic centimetres, so the conversion is a factor of 1,000.
Because volume scales with the cube of linear dimensions. Doubling the radius multiplies volume by two cubed, which is eight. The same principle applies to spheres and any shape scaled proportionally.
A cone holds exactly one third of the cylinder with the same base radius and height. This is why the cone formula is the cylinder formula divided by three.
By displacement: submerge it in water and measure the rise. The increase in water volume equals the object’s volume, which is the classic method attributed to Archimedes.