# Relation between half angle of numerical aperture and angle of view

The numerical aperture is defined as

$$\mathrm{NA} = n \sin{\theta}$$

where $$\\theta\$$ is the half-angle of the acceptance angle of the lens.

For a lens consisting of multiple elements, is there any relationship between $$\theta\$$ and the angle of view of the lens? Can the angle of view (calculated either by the triangle made from the focal length and imager size or focal length and aperture stop) be used as the half-angle?

θ is the half-angle of the lens' angle of view.

Angle of view via trigonometry:

The camera format measurements and a scientific calculator can be used to calculate the angles of view.

As an example: The full frame camera format measures 24mm height by 36mm length. The diagonal measure of this rectangle is 43.27mm.

The diagonal measure is computed using Pythagorean Theory --- Find the square root of the sum of the squares of the height and length.

Next, we construct a right triangle with a base divided by 2. Industry practice is to publish the diagonal angle of view. The diagonal angle of view is the largest of the three which are height, length, and diagonal. Seems odd but keep in mind that TV sets and computer monitors are sold by size which is the diagonal measure of screen size.

To compute the angle of view of the camera we construct a right triangle. The height of this triangle is the working focal length. Say we mount a 50mm lens on a full frame camera. We construct an imaginary triangle, focal length is its height. Its base is diagonal divided by 2 = 2 = 21.6mm

For this lash-up this right triangle base = 21.6mm height = 50mm. We divide height 21.6 by 50 = 0.43.

Using a scientific calculator or trig table, find the Arctan of 0.43 = 23.4°. This is one half the angle of view. The angle of view is 2X 23.4 = 46.7°.

Using this method, we compute the other two angles of view which are 27° Vertical and 40° Horizontal.

Try it for a 100mm lens mounted on a full frame. Diagonal angle 24° Vertical angle = 14° Horizontal 20°