I occasionally plug this into Wolfram Alpha:
a^2+b^2=c^2, a/b=16/9, c=27
Click the “approximate forms” solution to get the width and height (a and b) for a rectangle where you know the diagonal (c) and the ratio (16/9). a or b can be specified at the end instead of c if you know the width or height.
I most often use this when I need to get the physical width and height of a monitor panel that I know the diagonal size of (since nearly all monitors are advertised by their diagonal panel size). With that information and the resolution, you can figure out the physical DPI of the monitor. (Not to be confused with the effective DPI of the operating system, which is used for things like converting font points and ems to pixels, and is usually independent of the monitor’s size and resolution: 96 DPI for Windows, 72 DPI for Mac OS, and 75 or 100 DPI for X11 historically, though many Linux distros are preset to 96 DPI today.)