The radian unit

Mathematical skill ●●●○○

To measure how long the Roman Colosseum is around (circumference), the archaeologist can choose to walk all the way around it with a measuring wheel. Or she could just measure the distance across (diameter) and use the formula:¹

\text{circumference}=\pi\;\text{diameter}

In fact, she could do it even more quickly by simply measuring the distance to the centre (radius), since two of those distances give the diameter:

\text{circumference}=\pi \;\overbrace{2\;\text{radius}}^\text{diameter}

So the circumference (the full round) is ‘\pi-times-the-diameter long’ or in other words ‘\pi diameters long’² or ‘2\pi radiuses long’.³