 Okay, so we've recently seen in the previous video that the traditional area equals one half base times height. The area formula we see for triangles in Euclidean geometry doesn't work in hyperbolic geometry, it leads to a contradiction. It defects the contradictions with the defect function. So what can we do for area in hyperbolic geometries? How are we gonna interpret what area means? But honestly the answer is standing in front of us right now and then that's not just because of what you see on the slide right there. The conflict we had before is we chose an area function that was, that did one thing and then the defect was sort of saying another thing, right? There was this conflict, right? Between, because triangles which have equal area will be equivalent and we've already seen that equivalent triangles will have equal defects. So any area function hyperbolic geometry will have to be compatible with this defect function. And it turns out that's actually how we're gonna define area in hyperbolic geometry. The area of a hyperbolic triangle is just gonna be its defect. Cause after all, area functions have to be positive. And we've seen that the defect of a hyperbolic triangle is always positive because the angle sum is strictly less than 180 degrees. We have, we see that area functions have to be congruence preserving. Remember that means that congruent triangles have to have equal areas. Well, congruent triangles will have the same angle measures and therefore their defects are the same. So defect is congruence preserving and then area functions have to be additive. That is if you were to dissect a triangle or polygon into smaller triangles then you could add those areas together to give you the area of the original shape. Well, defect does that as well. Defect is positive, congruence preserving and additive. Since defect satisfies the three axioms of area it is an area function. And it turns out that in hyperbolic geometry that all area functions are just the defect of the triangle up to some coefficient K that pops up there. And so this number K here just like the Euclidean area formulas we saw this proportional constant there. The coefficient K should be understood as a unit conversion factor much like if we're converting from like acres to square miles or a square feet or something like that. And so most of the time we'll just take K to equal one because it's just trying to measure the area units we have in this discussion right here. So if you want to calculate the area of a hyperbolic triangle, let's say we have a triangle ABC like this. And let's say that the angle A measured to be 30 degrees and the measure of angle B turned out to be 60 degrees. And then let's say that the measure of angle C turned out to be 50 degrees or something like that. You'll notice when we add these things together what is the angle sum? The measure of angle A plus the measure of angle B plus the measure of angle C. Here we get 30 degrees plus 60 degrees plus 50 degrees. 30 and 60 give us 90 degrees plus another 40 degrees that gives us 150 degrees gives us 140 degrees. This is the angle sum of the triangle. When we calculate the defect of said triangle we then take 180 degrees take away 140. And so the defect of this is gonna be 40 degrees. And this is in fact the area of the triangle. The area of this triangle is 40 degrees. And this seems a little bit odd because you're like, wait, wait, wait, wait, wait. Area measuring with degrees? Degrees measure angles. They don't measure area. Well, that's in Euclidean geometry, Sam. If you want to come over to the hyperbolic world you measure area using angles, believe it or not. Cause you might be like, well, no, no, no I want to measure area with square inches. What about that? Well, let's pause there for a second Sherlock, square inches. What's a square? We're talking to citizens of hyperbolic land right now, right? There's no such thing as rectangles and therefore there is no such thing as squares. You can't measure area using square units in hyperbolic geometry cause there's no squares. You're gonna have to use a different measurement to measure area in hyperbolic geometry. And it turns out that angle measure is exactly how you do it. If you don't want to do degrees you can measure area in radians. That is kind of like the preferred method or if you're really bizarre you could try doing gradients or something. What the heck does that all mean? Anyways, we're going to measure area of hyperbolic triangles with these angle measures right here. And another thing I want to mention is that because defects sits between zero and 180 degrees likewise are areas of triangles also sit between zero and 180 degrees if we're doing degree measure or it's gonna be zero and two pi if we prefer radians. And so this tells us that triangles can only get they can't get arbitrarily large they're kind of bounded in terms of their size. If we look at the disc model of hyperbolic geometry if we have a teeny tiny little triangle like that you can barely see it how adorable. If you have a teeny tiny triangle its area is gonna be small its defect will be small that's what we mean by a small triangle. And if it has a small defect that means its angle measure is actually really close to 180 degrees. So on a small scale little triangles their angle sums will be really close to 180 degrees. So if you look at a very small triangle compared to the universe it lives in it's hard to tell the difference between a hyperbolic triangle and a Euclidean triangle because you can't really the defect is gonna look really close to zero. And that actually poses an astronomical issue for us here on planet earth right as we look at our universe do we live in a hyperbolic geometry or a Euclidean geometry? It's hard to say because if we measure triangles any triangle on earth is a small triangle compared to our universe. So we'd have to be looking at triangles that connect like galaxy clusters together in order to try to get any distance at any reasonable defect measurement. So that's gonna be a hard problem for our astronomers there to worry about but what if we look to something like this what if we took a hyperbolic triangle looks something like this now what I'm drawing right here is actually not a hyperbolic triangle. This is actually what's called an ideal triangle because I'm using as vertices of the triangle points at infinity which are not points inside of the geometry. You can't use them as in terms of geometry those three lines you see are actually parallel lines in the geometry they don't touch each other from this model it kind of looks like a triangle right and if you're to look at the angle sums here this is kind of what happens as the defect goes towards 180 degrees you're gonna get an object that looks something like this and so this ideal triangle again which is not a triangle it's area because it is a region in the plane it's gonna be 180 degrees and so triangles can't get this big because again we had to go towards infinity to accomplish this and so triangles believe it or not sit between zero and pi radians in terms of area again it's a really fascinating subject but hyperbolic geometry in fact does that stay tuned in the next video I wanna talk about how distances work here because although the area of this thing of triangles is bounded area can't get bigger than pi radians it turns out that the side lengths can get arbitrarily large right these lines have infinite length so how does one deal with length in the hyperbolic realm well stay tuned another cliffhanger bum bum bum