Two pendula of length "l," suspended from points separated by a horizontal distance a, can be joined in a trapezium configuration by linking the bobs with a light inextensible string of length b < a. Here we introduce this 'trapezoidal pendulum' as a charming and novel oscillatory system suitable for introductory physics laboratories. In particular, we show that the time period t is given by a scale-invariant equation of the form ([torque]/[torque][subscript s]) = P(a, b, l), where [torque][subscript s](l) = 2[pi][square root]"l/g" (with "g" gravity), and P(a, b, l) is a dimensionless function of a, b, and l. The trapezoidal pendulum exhibits richer behaviour than the canonical simple pendulum, but is nevertheless easy to construct from basic household items, making it an ideal system for home experimentation. Indeed, here we verify the [torque] dependencies empirically using modest, improvised apparatus. These investigations suggest exciting possibilities for distance teaching activities, such as collaborative studies of the dimensionless scalings, using data obtained by students working independently, and on apparatus of different sizes.