How to Calculate a Camera's Sensor or Movie Crop Factor

by Sean Setters
 
I was reading an article yesterday that stated the Canon EOS R had a "1.83x crop factor" when recording 4K video. However, while the EOS R certainly has a crop factor in 4K recording, it's actually a slightly lower crop factor of 1.75x.
 
I can understand the source of the confusion and, as such, I thought we'd take a minute to go over the math involved in determining a camera's sensor or video recording crop factor.
 
Before we dive into the crop factor calculations, it's important to understand why a crop factor is relevant. The crop factor determines the field of view we see in a given situation. For more information on the root of this phenomena and how it relates to sensor size, check out our Field of View Crop Factor explanation.
 
At the heart of it, the formula for determining crop factor is easy: you determine the ratio of the larger area's hypotenuse (diagonal) to the smaller area's hypotenuse measurement. For that, we'll need to use the tried-and-true Pythagorean theorem (a2+b2=c2). For example, to determine the crop factor of a Canon APS-C sensor, the math looks like this:
 
EOS-1-series Full-Frame Sensor Hypotenuse (mm)
362 + 242 = c2
1,296 + 576 = c2
1,872 = c2
√1,872 = c
43.27 = c
 
EOS 7D Mark II APS-C Sensor Hypotenuse (mm)
22.42 + 152 = c2
501.76 + 225.00 = c2
726.76 = c2
√726.76 = c
26.96 = c
 
Now the crop factor can be calculated by dividing the full-frame hypotenuse by the APS-C one:
 
APS-C Crop Factor
43.27 / 26.96 = 1.605
 
We know that APS-C sensor cameras feature a 1.6x crop factor, so with a little rounding, the calculation proves correct in determining the crop factor. The process for calculating video crop factor (when the video recorded is sampled via a 1-to-1 readout of the pixels in the center of the sensor which creates the video's resolution) is similar, but not exactly the same. The first difference involves our units of measure; we'll be using pixels to determine the hypotenuses for comparison. We can do this because the pixel size is the same (a standard) throughout the comparison. The second difference is that we'll need to normalize the aspect ratios to figure out the appropriate crop factor.
 
As I mentioned the EOS R above, we'll use it as an example of how to calculate the camera's (4K) video crop factor. First we need to determine what part of the sensor would be used if the video recording utilized the entire width of the sensor. To do that, we need to calculate the pixel area of a 16:9 (4K) ratio crop of the camera's 3:2 ratio frame. To do that, we simply divide the sensor's pixel width by 16 and then multiply by 9 to get the area utilized by an uncropped 16:9 video.
 
EOS R Video Pixel Height (Full Sensor WxH = 6720 x 4480 px)
6720 * 16 / 9 = 3780
 
So in a world where the EOS R captures "uncropped" 4K footage, the sensor would utilize an area of the frame that is 6720 x 3780 pixels and then downsample it to the lower 4K resolution. The video is technically cropped from a 3:2 ratio frame, but it's referred to as "uncropped" because the horizontal field of view remains the same (it's cropped at the top and bottom, not the sides).
 
Now that we know the portion of the frame that would be utilized for a 16:9 aspect ratio video, we need to compare it to the video size actually being recorded by the EOS R, or 3840 x 2160 pixels. Because of the rather large numbers, I'll shorten the following equations by eliminating some of the calculation steps.
 
"Uncropped" 16:9 Video Hypotenuse (px)
√(67202 + 37802) = c
7710.175 = c
 
EOS R 4K Video Hypotenuse (px)
√(38402 + 21602) = c
4405.814 = c
 
Crop factor of EOS R
7710.175 / 4405.814 = 1.75
 
So for practical purposes, the EOS R has a 4K crop factor of 1.75x, where a 20mm lens delivers a 16:9 field of view equivalent to a 35mm lens when recording in 4K. From a technical standpoint, saying that the EOS R has a 1.83x crop factor could be accurate if we also labeled the Sony a7 III as having a 1.05x crop factor in video mode. But when you call the a7 III's 4K video "uncropped," you necessitate normalizing the field of view captured in the EOS R's 4K 16:9 aspect ratio frame, resulting in the 1.75x result.
Posted: 10/5/2018 11:56:02 AM CT   Posted By: Sean
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