Real Area of a picture?

Question

I want to know the actual area of the entire area covered by an aerial photograph. _The Photo is 4000x3000 pixels _The Focal Length of the camera is 20mm. _The Picture height is 51 meters. I researched and got the scale of the photo. In this case it is 1 mm / 2550 mm. I understand the scale tells me that if I move 1 mm inside of the photo , in the real terrain I would be moving 2550 mm. I want to know how to get the real width and height of the picture. ¿? cm. I need this data to know the area of the photo and therefore know the terrain area that is pictured. Thank you for your help.

Answer 1

This is a geometry question, but the math is simple. The area of a rectangle is the width X the length. [NOTE: I use "length" to refer to the longest dimension of the rectangle, and "width," the shortest dimension.] Measure the width and length of your photo and multiply the width times the length to get the area of the photo. Now, to scale it up to the actual terrain, multiply the width of the photo times 2550 and the length of the photo times 2550. Take the results of these two operations and multiply them together to get the area of the terrain. For example, if the photo is 200 mm wide and 300 mm long, multiply 200 X 300 to get the area of the photo. The result is 60,000 square mm. For the larger area (that of the actual terrain), let's translate to meters. 1000 millimeters = 1 meter. Divide 200 by 1000 and 300 by 1000 to get the photo's width and length in meters. Result = .2 meters wide by .3 meters long. Now multiply these times 2550 to scale up to the terrain. Results: the width is 510 meters and the length is 765 meters. (This is a large area: 1000 meters is a kilometer which is .62 of a mile). So, to calculate the area of the terrain you have photographed, multiply 765 X 510 for a result of 390,150 square meters. This is equal to 39 square kilometers or to about 15 square miles. The city of Seattle is about 84 square miles, so you have photographed almost one quarter the area of Seattle! If by "Picture height is 51 meters" you mean your 20 mm camera lens was only 51 meters (about 165 feet) above the ground, it seems to me you would not have been able to photograph one-quarter the area of Seattle. So, either my math is wrong (and you should ask this question to a math tutor) or your research on the scale (1:2550) is off. I have used the formulas on this page: https://photo.stackexchange.com/questions/3573/estimating-focal-length-range-required-for-shooting-scenario. The result I get is that the length of the terrain you photographed from 51 meters above ground using a 20 mm lens (and assuming you have a full-frame sensor) is 92 meters (299 feet). If your sensor is 36 mm by 24 mm (standard full-frame sensor), we can calculate the width of your scene as it is proportional to the dimensions of your sensor. Result: width is 61 meters (199 feet). This gives us an area of 92 X 61 = 5,643 square meters or .0056 square kilometers, much more like what I'd expect. Let me know if this helps.