How the angles relate
Half the horizontal angle gives a tangent. Dividing that by the aspect ratio gives the tangent of half the vertical angle, and doubling the arctangent returns the vertical angle.
The diagonal uses the same two tangents as the legs of a right triangle: the square root of their squares gives the tangent of half the diagonal angle.
Worked example
A 90 degree horizontal field of view at 16:9 has a tangent of 1 at half angle. Dividing by 1.7778 gives 0.5625, so the vertical angle is 58.72 degrees and the diagonal is 97.85 degrees.
Half the vertical angle is 29.36 degrees. Note that 90 plus 58.72 is not 97.85: angles do not add across axes.
Why the same FOV looks different
A wider aspect ratio at the same horizontal angle gives a shorter vertical angle, so you see less above and below. Games often fix the vertical angle instead, which widens the horizontal view on ultrawide screens.
That is why a setting labelled 90 can show a different amount of the world on two monitors.
Limits of this calculator
It assumes a rectilinear projection. Fisheye, equirectangular and anamorphic projections do not follow this relationship, and lens distortion changes the effective angle.
It does not model the difference between the rendered field of view and the visible field on a curved screen.
For the ratio itself, use the aspect ratio calculator.
Inputs for the FOV calculator
Horizontal field of view
The horizontal angle your game or lens is set to, in degrees.
Aspect ratio width
The first number of the ratio, such as 16 in 16:9.
Aspect ratio height
The second number of the ratio, such as 9 in 16:9.
CALCULATOR QUESTIONS
FOV calculator questions
Do the angles add up?
No. Field of view angles describe a pyramid, so the diagonal is not the horizontal plus the vertical.
Why does my game look wider on an ultrawide?
If the vertical angle is fixed, a wider screen widens the horizontal angle and shows more of the scene at the sides.
Is this the same as a camera lens angle?
It is the same geometry, but a lens is quoted diagonally and includes distortion that this model ignores.
What about 21:9?
Enter 64 and 27 for the true ultrawide ratio, or 21 and 9 for the marketing label.