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How to calculate sensor size (width/height) when knowing focus length, vertical & horizontal angle view (FOV)?

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We can dash out the path of image forming rays as they travel from lens to imaging chip. They trace out a triangle. The apex of this triangle has its origin at a point called the rear nodal of the lens array. The angle of this trace is the same as the published angle of view.

If we know the focal length of this lens, we know the distance rear nodal to the imaging chip. This is the height of this triangle.

Say the focal length of the taking lens is 4mm. This dimension is the height of the triangle.

Say the angle of view is published as 50⁰. This is the angle at the apex of this triangle trace.

The math is easier if we work with a right triangle. To accomplish we bisect this angle thus 50⁰ divide 2 = 25⁰. The apex of our right triangle is 25⁰. The height of this triangle remains 4mm.

Now that we have a right triangle we can easily use the trigonometric function Tan. Tan is the ratio of the length of the opposite side of the triangle over the adjacent side. We consult a trig table to find the Tan of 25⁰. The table tells us Tan 25⁰ = 0.4663.

We want to find the length of the base of this triangle. One half the base will be: 0.4663 X 4 = 1.8652mm. The length of the base of the entire image triangle is 2 X this value thus the base dimension is 2 X 1.8652mm = 3.73mm.

We do this math for both the horizontal and the vertical angle of view and we have calculated the chip dimensions.

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