Hi there, and welcome to yet another installation of Weird Science, a series of blog entries here on Musings where I take a look at the fictional and apply scientific and mathematical fact to it. Today, I'm headed into the forest lovely, dark, and deep to tackle the trigonometry of the Slenderman. Care to join me, reader? Yes? Well, alright - just make sure you stay close! Stragglers never fare well in these woods, especially when it's night time and they're all alone...
Now, as my more loyal Muselings will recall, I love discussing mathematics and science. Now, usually that pertains to statistics and chemistry, but what some of you might not know is that I'm pretty good at trigonometry, too. I can judge angles by looking at them. I can solve for the sine of theta and the number of radians in any given degree with the best of them, and I fully believe that just about anything measurable can be solved with a little trig. So today, I'm going to apply that trig knowledge to a favorite horror fandom of mine, and specifically a scene in my short Slenderfic The Hunted. For those who haven't read it yet, this blog entry is spoiler-free, but does discuss the story's beginning. If you'd like to read The Hunted for context before you read this blog entry, you can do so by clicking right here, but no context is needed to understand what I'm doing here.
Let's start with the scene from the story in question - my protagonist, Jeremy, up in his hunting blind which is eight feet off the ground, looks out of it to see Slendy standing about 30 feet away from him. He is able to look the being in the eyes - or he would be able to, anyway, if Slendy had any eyes to look into. Jer, confused and more than a bit startled at the apparent height of the being, looks down at the ground at an angle of 17° from horizontal to double-check what he's seeing. The question is, can he estimate how tall Slendy really is by doing this?
Before we answer that question, let me give you guys a little right triangle trig review, since we'll basically be working with right triangles anyway:
The right triangle has three parts - an adjacent side (sometimes called b), an opposite side (sometimes called a), and a hypotenuse (sometimes called c). You can find one side if you have the other two by using the Pythagorean theorem, which states that a^2 + b^2 = c^2. That funky-looking little O with a line through it is the Greek letter Theta, and it denotes an angle that isn't a right angle. Hopefully, you remember that the square in the corner means a right angle (which measures 90°;), while the arc next to theta means any other angle. You can find the a missing side if you have only an angle and a side by taking the sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), or tangent (opposite/adjacent) of the angle in question, depending on what side it is you want to find. You can find a missing angle by taking 180°, and subtracting the other two angles you do have from it.
Now onto the question at hand. Could Jeremy accurately measure Slendy's height with trigonometry based on what he observes around him? The answer is yes, of course - but how?
Well, let's look at what we know. First off, we know that Slenderman is going to be be making a right angle with both the ground and its line of sight to Jeremy, since it's standing 30 feet from Jeremy and is also standing straight up and down, as Slendermen are wont to do. Jeremy can also look down at an angle of 17° to see its feet. This makes a triangle with an unknown opposite side (Slendy's height), an adjacent side 30 ft long, and an unknown hypotenuse that makes an angle of 17° from Slendy's feet to Jeremy's position 30 feet away. We'll call this triangle the SlenderTriangle. Note that the SlenderTriangle is missing two sides, but we can fix that later.
As for Jeremy, his story's a bit different. We know that Jeremy's hunting blind makes a right angle with the ground, since the tree grows straight up, and we know he can look down at the ground at an angle of 17° when he's looking at something 30 feet away from him - in this case, Slendy's feet, or more likely, the foliage and trees obscuring its feet, since this is in a forest. This creates a triangle with an adjacent side of 30 feet and an opposite side of 8 feet, with an unknown hypotenuse between Jer and the ground that forms an unknown angle between Jeremy's position and the ground 30 feet away. We'll find this angle later - we can't definitively give the angle as 17°, because in the story Jer only says he can look Slendy in the non-face, not that Slendy is any specific height. That means the SlenderTriangle may not be a right one, depending on Slendy's actual height, but we'll assume it is one. As for Jer's Triangle, as we'll call it, we do know it's a right triangle and thus can apply right triangle trig rules to it. Notice that both our triangles need the same hypotenuse.
Since Jer's Triangle has two sides, we can use the Pythagorean Theorem to get the hypotenuse. Because the SlenderTriangle's not completely right, we can't do this trick with it. So, setting up the Theorem for Jer's Triangle, we get 30^2+8^2=c^2. After that it's a matter of simple number-crunching to get our hypotenuse, which turns out to be 31.04 feet. We now have all three sides of Jer's Triangle and two sides of the SlenderTriangle. We can now estimate how tall Slendy would be, both with the information we found and the information given.
Now, according to the very, very loose canon surrounding Slenderman and its Mythos, the entity in question can be anywhere from 6 to 12 feet in height at a given time, and can adjust its height at will. If we take a mean height of this range, we get 9 feet as the average height for our faceless friend, which is pretty accurate to how most people tend to depict it. So we're hopefully aiming for a height of 9 feet for the SlenderTriangle's opposite side, but a bit taller or shorter would also work, although hopefully not below 8 feet (too short for the blind's height) or above 10 feet (too tall for the blind's height).
Given that range to aim for, let's look at Slendy's height. Since we have an angle and two sides, we could use the Pythagorean Theorem to solve for the opposite side. We could also use the Law of Sines if we assume the triangle's not a right triangle, but that's a bit too convoluted. So, assuming the SlenderTriangle is a right one, we can simply adjust the equation to find sine a bit and get our answer. Currently, if we stuck all the numbers in, we'd get the following: sin(17°) = opposite/31.04. So to solve for the adjacent side, all we have to do is move our hypotenuse value over. That gives us an equation that looks like 31.04 * sin(17°) = opposite, and if we solve this equation, we sure enough get a value of 9.08 feet for the opposite side. That means Slendy is, in this scenario, standing just a bit over its average height of 9 feet. An intimidating height for it to stand at to be sure, and definitely reason for poor Jeremy to have pause when he looks out of his blind and sees it standing there!
This can also work the other way. If we wanted to give Slendy a mean height of 9 feet and measure the distance from it to Jeremy, we could take the Pythagorean Theorem and solve for the adjacent side of the SlenderTriangle, or we could multiply our hypotenuse by the cosine of our 17° angle. Either way, we'd get the same result - a 29.7-foot distance between Tall and Scary and our hapless hunter, Jeremy. This value is off by slightly less than a full foot, but it's still close enough to the 30-foot distance of Jer's Triangle to work for the proportions given. This is still assuming that the SlenderTriangle is a right one; if not and the adjacent side really is 30 feet, we'd need to use the Law of Sines to solve this - right triangle trig wouldn't work.
Now there's just one more question to solve, since I promised I would - what is the exact angle that occurs between Slendy's feet and the hypotenuse, since Jer could also look down at them to help him calculate his suit-wearing stalker's height? To solve this, we actually need to use the tangent formula, which is found, as you recall, by dividing the opposite side by the adjacent side. Actually, we could use any right triangle trig formula we wanted to find this angle, but using the tangent is easier since we could have done that calculation before we got the hypotenuse anyway. So, following that calculation, tan θ = 8/30, but we need the angle, not the actual tangent of said angle. That's where the inverse of tangent comes in, which here means that arctan(opposite/adjacent) = θ. Therefore, if we plug in our numbers and solve, we find that θ = 15°. Well, actually, it more accurately equals 14.9°, but the value's close enough that you could still call it 15° and be safe. However, if you were to use that angle in a calculation here, you'd probably want to play it safe and go with the more accurate value.
That's it for this installment of Weird Science. Hope you had a fun, safe trip through the forest, and - hey, did we lose a few back there? Oh... guess not. I can see them up in those trees over there. They look kind of dead, though, judging by the branches sticking through their torsos... what do you say we get out of here and leave well enough alone? I'd hate to lose any more valued readers...