The image sensor size in compact cameras is given as some fraction of inch. For example, 2/3". This is only a rough guideline and does not reflect the real size in any dimension - whether horizontal, vertical, diagonal, inch or metric.
How this number relates to the physical sensor size is something I wondered about for years. From other references there appears to be a historical connection to real sizes... Something to do with old vidicon tubes, external vs. internal dimensions, glass envelope, etc. Wiki gives a comprehensive table here: Table of sensor formats and sizes. It does not go on to explain the correlation, if any, between the odd sensor spec and the actual dimensions.
Below is a table which does just that. The key, I found, is in the third column. Pretend there are 18 millimetres per inch for sensors below those designated 1/2" and 16 millimetres per inch for those above. That way we can obtain their true diagonal dimension. The width and height figures can be extracted from that once we include the aspect ratio.
The final two columns crop and stops show the crop factor and the number of stops that the sensor is below the light gathering ability vs. "full size" sensors.
From an idea March, 2013.
The APS-C and full-sized sensors are included only for comparison.
To obtain the width (W) we apply the following formula. D is diagonal and A is aspect ratio.
The width and height of the sensors matches the official measurements for most sensors. The only exception is the 1.5" sensor which should probably be designated 1.46" to match the numbers. Therefore my "reverse engineering" of the sensor size designation is valid.
As long as one can remember at which point they change from 18mm per inch to 16mm per inch, and what the aspect ratio is, one can obtain accurate width and height measurements from the published sensor size.
Revised 11 Apr 2015