The President, the Shotgun and the Hypotenuse


I am actually writing this post to talk more about math then the presidential Skeet Photo or Skeetgate ,Skeeters or whatever manufactured crisis you want to call this. As I have written before I shot competition pistol during High School so I am family with pistols and firearms in general but my shotgun experience is limited. I could say that 100% of my experience shooting shotguns involved trap shooting and that would sound impressive. That experience amounts to eight rounds fired out of a 20 gauge during Boy Scout camp about twenty years ago. I did hit all eight pigeons without a single miss in that session but I liked shooting pistols or rifles more. Plus shotguns are so much more expensive then small caliber pistols or rifles. At the time it was 25 cents per shell to try out the shotguns at camp. I had a dollar so I got four shots but after I hit all four the scout master manning the range gave me four more as a freebie to see if I could repeat my performance.

I used hypotenuse in the title because I thought it sounded better and because of The Lion, the Witch and the Wardrobe reference. I did not actually calculate the hypotenuse but you could from the information in this picture some guesstimation and basic trigonometry. I always think it is interesting when you can insert math into popular culture or current events. Math is terrible important and it actually seems to frighten people. That being said do not assume I am some kind of math wiz. I left engineering and switched majors after I failed differential equations and an engineering specific statistics class. I hit a wall in math that I just could not seem to get past but I did do well in future non-engineering related statistics classes.

What got me thinking about trigonometry was the claim by some people when looking at the photo that the Obama is aiming the shotgun too low to hit a clay pigeon target in Skeet. It looks like he is aiming the shotgun at just about level ground. I cannot comment on his technique in regards to how he is holding the gun or stance because I do not know enough about shotgun shooting. He does look like he is holding the gun a little high and close to his cheek but then he is also a lefty which really screwed up the proportions to me when looking at shooting. I can shoot right or left handed with a pistol with near equal accuracy but I cannot shoot a rifle or shotgun from the left side at all. I just have no frame of reference. What I can comment on is the math.

On the low resolution pictures published in most news organizations the shotgun does appear to be nearly level. Go to the high resolution original though and you can see this is not the case. The angle is very small but it is there. It is less then ten degrees but more then zero. I decided to split the difference and estimated it at somewhere around five degrees before doing any calculations.

When shooting skeet typical ranges are around twenty-one yards or sixty- three feet but can be more or less depending on your shooting position and where you engage the target.


Here is a handy diagram I found to help with visualization. There is a great description of the origins of Skeet and how the modern range is set up right here.

Notice the target cross point at twenty-one yards or 63 feet. That is important and what is also important is the low house fires clays three feet six inches above station seven. My initial estimation of the angle the shotgun is being held at from the picture is five degrees. I had to remember SOHCAHTOA but then it was easy. We know the distance across the ground, not the exact distance but an example potential distance, and we have an estimate of an angle.


Now just solve for the h or height.


h= 5.51 feet rounded to the nearest hundredth of a foot.

Then add in the approximate height that Obama is shooting from. I do not know how tall Obama is so I had to look it up. He is six foot one which is taller then I thought. Anyway according to this mean segment length of an adult male head and neck is 10.75 percent which would put Obama shooting from 65.15 inches off the ground. Add that into the calculated height of our triangle and you have the theoretical point he was aiming at assuming a 63 foot length which is 10.94 feet off the ground. If he was aiming at a target that is farther away then the height would be more and conversely a target closer to him would be lower. You can calculate your own example distances using the above formula and if you wanted to get real fancy you could do something in excel and make a graph.

After doing this estimation I thought to try and calculate an angle by overlaying some triangles onto the picture itself.


I measured a hypotenuse of 12.3 cm and a height of 1.5 cm for the triangle made up by the shotgun to roughly Obama’s eye line. By using the sin function you can calculate the angle of fire.



To calculate the angle we need to use the inverse sin function which gives you an angle of about seven degrees.

Going back to the tangent function an angle of seven degrees equals a height of 7.74 feet plus the height he was shooting from which we have already calculated so a total of 13.17 feet. Either of those aiming points are theoretically possible on the skeet range where the clays are required by regulation to cross each other at a height of 15 feet.

I could of course be off in my calculations or estimations by trying to go off a picture but it is obvious that he is not holding the shotgun perfectly level and it is also obvious that he is not holding the shotgun at an angle above ten degrees. I can only be off by so much.

A ten degree angle would equal a aiming point of 16.54 feet and a angle of one would be an aiming point of 6.53 feet. This is still assuming the target is 63 feet away horizontally. Neither of those angles sound very plausible based on eyeballing the photo or calculating the angle based off the photo.

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