How can I calculate a camera sensor's width and height?

It is not possible to calculate the sensor size with that information because the 1/3" measurement refers to the diameter of a hypothetical tube that would enclose the sensor. That sets a limit on the diagonal of the sensor but it not an exact fit.

The the sure way to know the sensor-size is to measure it or find the part number and look at specifications of that particular sensor from the manufacturer.

There are common approximations and Wikipedia has 4.8 x 3.6mm for a 1/3" sensor with 4:3 aspect-ratio but you will need to figure out if that matches your sensor and what region was cropped to get the 16:9 aspect-ratio of your images. Sometimes, only the vertical dimension is cropped but sometimes both, effectively creating an equivalent of a smaller sensor. There are native 16:9 aspect-ratio sensors too, and they can fit into 1/3" but that could be wider than 4.8mm.