Calculating Field of View with Various Focal Lengths

    Choosing the right lens for an application can be difficult.  To make matters worse lens sizes are typically measured in terms of focal length, which is quite abstract when is comes to real world use.  Within this article, we explain how focal length correlates to the field-of-view in a simplified manner.

Key Definitions:

• Field of View - The angular measurement of the scene being captured.  Measured in degrees, the field of view can be diagonal, horizontal, or vertical.
   
• Image Sensor - The sensor used to capture the image.  The sizes of image sensors are measured in inches diagonally.
   
• Focal Length - A distance measurement correlating the light convergent power of the lens.  The distance is measured in millimeters from the lens to the image sensor.  The longer the focal length the less convergent power creating more of a “zoom” effect.  The shorter the focal length the more convergent power creating more of a “wide angle” effect.

    The relationship between focal length and field of view is quite simple; a shorter focal length portrays a larger field of view, and a longer focal length will result in a smaller field of view. Using focal length and the image sensor size, which are usually supplied by the manufacturer, we can calculate the exact field of view that will be captured.  To do this we need some basic trigonometry.

Note: The following assumes that image sensor size, matches the image size output of the lens.  This is the case with most bullet and purpose built cameras.  This is not the case with some cameras that are designed to work with standard film lenses, where the image output of the lens is larger than the image sensor itself.

Formula Definitions:

• Vertical Angles - angles opposite of each other when two lines cross.
   
• Tangent - Function used to determine the ratio of the opposite and adjacent sides of a right triangle based on the acute angle.

  tanθ = (b/a)
  b = opposite side
  a = adjacent side
  θ = angle
   

• Arctangent - The inverse function of tangent.  Used to calculate the acute angle within a right triangle given the opposite and adjacent sides.

  arctan(b/a) = θ
  b = opposite side
  a = adjacent side
  θ = angle

    These trigonometric properties allow us to easily calculate the field of view from a know focal length and image sensor size.  If you were to create two lines from the respective corners of the image sensor and have them intersect at the principal point of the lens, this would create a triangle by which we could calculate the field of view.  The principal point of the lens will be one focal length from the image sensor.  The focal length is shown by “a”, and the image sensor size is shown as 2*b.  This will create two identical right triangles, allowing us to use the “arctan” function.  Field of view can be determined from θ.  From the diagram we see 2*θ is the total angle, since line “a” is a bisector.  Using the property of vertical angles as defined above, 2*θ is also the identical angle of field of view.

Let’s try a couple examples.  Start with the known image sensor size and focal length values.

NOTE: For increased accuracy use at least five significant digits.

Example 1:
    Image Sensor Size: 1/3"
    Focal Length: 2.5mm
    Field of View: 2*θ, where θ is unknown

    Step 1: Convert to similar units.
                    1/3" = 8.467mm

    Step 2: Assign Values.
                    a = 2.5mm
                    2b = 8.467
                    b = 4.233 (2b divided by 2)

    Step 3: Substitute Variables.
                    arctan(b/a) = θ
                    arctan(4.233/2.5) = θ

    Step 4: Calculate.
                    No calculator needed, just type the equation into Google.
                    arctan(4.233/2.5) = 1.037 radians (Google defaults to radians)
                    θ = 1.037 radians

    Step 5: Convert radians to degrees.
                Just type the conversion into Google.
                1.037 radians to degrees
                1.037 radians = 59.416 degrees
                θ = 59.416 degrees

    Step 6: Calculate Field of View
                FOV = 2*θ
                FOV = 2*(59.416) = 119.832 degrees

Example 2:
    Image Sensor Size: 1/3"
    Focal Length: 16mm
    Field of View: 2*θ, where θ is unknown

    Step 1: Convert to similar units.
                    1/3" = 8.467mm

    Step 2: Assign Values.
                    a = 16mm
                    2b = 8.467
                    b = 4.233 (2b divided by 2)

    Step 3: Substitute Variables.
                    arctan(b/a) = θ
                    arctan(4.233/16) = θ

    Step 4: Calculate.
                    No calculator needed, just type the equation into Google.
                    arctan(4.233/16) = 0.258 radians (Google defaults to radians)
                    θ = 0.258 radians

    Step 5: Convert radians to degrees.
                Just type the conversion into Google.
                0.258 radians to degrees
                0.258 radians = 14.78 degrees
                θ = 14.78 degrees

    Step 6: Calculate Field of View
                FOV = 2*θ
                FOV = 2*(14.78) = 29.56 degrees

    Because the image sensor measurement is diagonal the examples listed above calculate a diagonal field of view.  To calculate horizontal or vertical field of view we will need another conversion.  Using Pythagorean Theorem we can use our now acquired diagonal FOV measurement, combined with known aspect ratios to calculate the vertical and horizontal field of view.

Formula Definitions:

• Pythagorean Theorem - a^2 + b^2 = c^2

Using known aspect ratios we can express b in terms of a.
    If the aspect ratio is 4:3 then b = 3/4a
    If the aspect ratio is 16:9 then b = 9/16a

Let's try an example using a value we found above.

Example 1:
    Aspect Ratio: 4:3
    Field of View: ~120 degrees

    Step 1: Assign Values.
                    a = a
                    b = 3/4a
                    c = 120

    Step 2: Substitute.
                    a^2 + (3/4a)^2 = 120^2

    Step 3: Simplify.
                    a^2 + 9/16(a)^2 = 14400

    Step 4: Combine like terms.
                    25/16(a)^2 = 14400

    Step 5: Simplify.
                a^2 = 9216

    Step 6: Simplify.
                a = 96

    Step 7: Simplify.
                a = 96 degrees horizontal field of view
                b = 3/4a = 72 degrees vertical field of view

Using the above formulas you should have to tools necessary to decipher common focal length measurements.  The chart below contains common image sensor sizes, focal lengths and their field of views.