Angle of View and Dimension of Angular Field:
As you have discovered, our cameras sport a rectangular format. We can trace imaginary lines outward from our camera lens to determine the angle of view and the dimensions of the angular field. Such a trace reveals that the pyramid has a rectangular base. The size of this base expands with distance. The trace of the sides of this pyramid reveals that each of the four sides is in the shape of an iisosceles triangle.
We can use the trigonometry of right triangles to most easily solve. If we bisect an iisosceles triangle, we create two “right” triangles. A “right” triangle has one angle that equals 90°. Now we can use the trigonometric functions of “right” triangles (the easiest trig) and solve for the size angles and the dimensions of the base. Our solution will be for ½ the triangle; we multiply by 2 to discover these values for our iisosceles triangle.
For the angle of view – solve for horizontal angle of view APS-C format 24mm long with 30mm lens mounted.
Angle of view = (ArcTan (d÷ f÷ 2))X2
d = dimension
f = focal length
Angle of View = (ArcTan (24÷30÷2) X 2
Angle of View = (ArcTan (0.40) X 2
Angle of View = 21.8 X 2
Angle of View = 43.6°
Solve for vertical Angle of View
Angle of View = (ArcTan (16÷30÷2) X 2
Angle of View = (ArcTan (0.27) X 2
Angle of View = 14.9 X 2
Angle of View = 29.9°
Solve for the dimensions of the Angular Field.
We bisect the angle to construct a “right” triangle from the side of the pyramid which is an iisosceles triangle. The height of this triangle is the distance lens to subject. The Tan function of the apex angle multiplied by the Tan value is the dimension sought.
Distance = 5 meters
Lens mounted is 30mm
For Angular Field Vertical
Divide 29.9° by 2 = 14.95°
Tan 14.95 = 0.267
Multiply Tan by distance = 0.267 X 5 = 1.335
This is ½ of Vertical field thus 1.335 X 2 = 2.67 meters = 2,670mm = 105 inches
For Angular Field Horizontal
Divide 43.6° by 2 = 21.8°
Tan 21.8 = 0.40
Multiply Tan by distance = 0.40 X 5 = 1.99
This is ½ of Vertical field thus 1.99 X 2 = 3.99 meters = 3,999mm = 157 inches
For any other distance, multiply tan of ½ the angle by distance and then multiply by 2. Answer will be in the same unit of measure as used for distance value.