The angle of the image or the angular field of the lens is the angle formed by the rays connecting the extreme opposite points of the frame with the optical center of the lens. In other words, this is the maximum angular size of an object that can be captured using this lens.
The wide angle of the image allows the lens to cover more space due to the small scale of the image. The narrow angle of the image shows less space, but on a larger scale.
Since the frame has a rectangular shape, it is necessary to distinguish the angular field, measured horizontally, vertically and diagonally. In the technical characteristics of photographic lenses, the largest, i.e. diagonal angle of the image.
The magnitude of the angular field is inversely proportional to the focal length of the lens and directly proportional to the size of the photosensitive material (film or matrix), i.e. the longer the lens and the smaller the matrix, the smaller the image angle, and vice versa, the shorter the lens and the larger the matrix, the larger the image angle.
Lenses with an image angle of 40-60 ° are considered normal or standard. If the angle of the image is greater than 60 °, the lens is short-focus or wide-angle, and if the angle is less than 40 ° – long-focus or telephoto.
How to find out the angle of the image for a particular lens? This is not difficult. Below you can familiarize yourself with the formulas for calculating the angular field of the lens, and if the mathematical details are not too interesting for you, you have the opportunity to immediately go to an interactive calculator that can perform all the calculations for you.
Image angle calculation
To find the angle of the image, it is enough to know the focal length of the lens and the linear dimensions of the matrix. The image angle is calculated by the formula:
α is the image angle (angular field) in radians;
d is the distance between the extreme points of the frame (width, height or diagonal) in millimeters;
f is the focal length of the lens in millimeters.
As you can see, school trigonometry can really be useful in life.
For example, we find the diagonal angle of the image for a standard lens with a focal length of 50 mm mounted on a full-frame camera. The dimensions of the full frame are 36 × 24 mm. Through the Pythagorean theorem we find the diagonal of the frame:
Substitute the length of the diagonal and the focal length in the formula of the angular field and get:
To translate the answer from radians to degrees, just multiply it by 180 ° / π (roughly speaking, one radian contains about 57.3 degrees). Thus, the image angle will be 46.8 °.
Knowing the angular field of the lens, you can calculate the maximum linear size of an object that fits into the frame. Obviously, in contrast to the angular field, the linear coverage of space directly depends on the distance to the object. The following formula is used to calculate the linear field:
D = 2 • R • tg (α / 2), where
D – coverage of space (linear field),
R is the distance to the object.
For example, at a distance of 10 m, the coverage of the space for the already mentioned 50 mm lens on a full-frame camera will be equal to:
2 • 10 • tg (46.8 / 2) ≈ 8.7 m.
In order not to waste time on all these trigonometric calculations, you can use a special calculator, but first I have to make one important reservation.
Focus and focal length
The above formula of the angular field assumes that the lens is focused on infinity. Only in this case, the effective focal length of the lens corresponds to the nominal. When focusing the lens on closer objects, the effective focal length can vary within certain limits, which entails a proportional change in the viewing angle. In most cases, swimming the focal length is very insignificant and can be safely neglected, however, in macro photography, when the distance to the object is comparable to the focal length of the lens, the effect of changing the viewing angle can become quite obvious.
Ideally, we should substitute the value of the effective focal length for each particular focusing distance into the formula, but, unfortunately, this is not always possible.
The calculation of the effective focal length of the lens is relatively simple and straightforward only for classic fixes, which are focused by moving the entire optical unit forward. In other words, their effective focal length increases as the shooting scale increases, and the viewing angle decreases accordingly. The effective focal length in this case can be found by the formula:
F is the effective focal length;
f is the nominal focal length;
R is the focusing distance.