Welcome to another Mathologer video. The shoelace formula is a super simple way

to calculate the exact area inside any convoluted curve made up of straight

line segments, like my cat head curve over there. Even the great mathematician

Carl Friedrich Gauss was impressed by this formula and mentioned it in his

writings. The formula was certainly not invented by him, however it's often also

referred to as Gauss's area formula, probably because a lot of people first

learned about it from Gauss (and not because someone calculated Gauss's area

with it :) In today's video I'll show you how and why this formula works. The

visual proof I'll show you is just as pretty as the formula itself and along

the way I can promise you a couple of very satisfying AHA moments to make your

day. I've got a special treat for you at the

end of the video: a simple way to morph the shoelace formula into a very famous

and very powerful integral formula for calculating the area enclosed by really

complicated curvy curves, like for example this deltoid rolling curve here.

Now obviously we call this crazy formula the shoelace formula because it

reminds us of the usual crisscross way of lacing shoes. Now let's make sense of

the shoelace formula and use it to calculate the orange area. I start by

filling in the coordinates of the blue points. Take one of these points and move

its coordinates to the right. Now we traverse the curve in the

counterclockwise direction and do the same for the other blue points we come

across. Here, there, there, there. Now we're back at the point we started from and

include its coordinates one more time at the end of our list. Now draw in the

crosses. Okay this green segment stands for the product of the two numbers at

its ends. So 4 times 1 equals 4.This red segment stands for minus the

product of the number at its two ends. So 4 times 0 equals

0. Minus that is - 0. Oh, well obviously the "minus" is not important here but it will

be later. Green again. So 0 times 5 equals 0. Red again, we need to

calculate minus the product, so 1 times - 2 equals -2. Minus that, and so on. So we get two

products for every cross, one taken positive and one negative. Now adding up

all the numbers gives 110. Okay, almost there. The formula tells us to divide by two.

So half of 110 is 55, and that's the area of my cat head. Really pretty and super

simple to use. And this works for any closed curve in the xy-plane no matter

how complicated. The only thing you have to make sure of is that the curve does

not intersect itself like this fish curve here. And it will become clear later

on why you have to be careful in this respect. Okay now for the really

interesting bit, the explanation why the shoelace formula works. It turns out that

the individual crosses in the formula correspond to these triangles which

cover the whole shape. Note that all these triangles have the point (0,0) in

common. Okay, so the area of the first triangle

here is just 1/2 times the first cross. So, again, the first cross is equal to 4

times 1 minus 4 times 0 equals 4, and half that is 2. And it's actually easy to

check that this is true using the good old 1/2 base times height area formula

for triangles. Now the area of the second triangle is 1/2 times the second

cross, and so on. But why is the area of one of these triangles equal to 1/2

times the corresponding cross? Here's a nice, really really nice visual argument

due to the famous mathematician Solomon Golomb. What we want to convince

ourselves of is this. So let's calculate the area of this triangle

from scratch. Actually what we'll do is to calculate the area of this

parallelogram here whose area is double that of the triangle. Okay let's start

with the special rectangle here. Then the coordinates translate into the side

lengths of these two triangles. First (a,b) turns into these two side lengths, and

then (c,d) into these. Color in the remainder of the rectangle and shift

the green triangles like this, and like that,

Now do you see the second small rectangle materializing? Right there. The

two triangles overlap in the dark green area and so we can pull the

colored bits apart so that they fill exactly the parallelogram and the little

rectangle. Since we started out with the colored bits filling a large rectangle

this means that "large rectangle area" equals "parallelogram area" plus "small

rectangle area". But now the areas of the rectangles are ad and bc. That's almost

it.

Now, without any words... Pure magic, right? And, of course, all of you who are

familiar with vectors and matrices will realize that another way of expressing

what we just proved is the mega famous result from elementary linear algebra

that the area of the parallelogram spanned by the two vectors (a,b) and (c,d) is equal

to the determinant of the 2 x 2 matrix a,b,c,d. Anyway, back to the shoeless formula.

At this point we just need to divide by 2 to get the area of the triangle and

that's it, right? That completes the proof that the shoeless formula will always work,

right? Well, not quite. We are still missing one very important

very magical step. Let's have another look at my cat hat, but let's shift it so

that the point (0,0) is no longer inside and again move around the curve and

highlight the triangles whose area the shoeless formula adds. This time let's

start here. As we move around the curve in the counterclockwise direction the

green radius which chases us also rotates around (0,0) in the

counterclockwise direction. Something does not look right here. The yellow

triangles are sticking out of the cat head and at this point the combined area

of the triangle is larger than that of the cat head and should get even larger

as we keep going. However, whereas up to now the radius has been rotating in the

counterclockwise direction, at this point it starts rotating in the clockwise

direction and this change in sweeping direction has the effect that the

shoeless formula subtracts the areas of the blue triangles. And this means that

the area calculated by the shoelace formula will be the total area of the

yellow triangles minus that of the blue triangles which is exactly the area of

our cat head again. The same sort of nifty canceling of areas makes sure that

no matter how convoluted a closed curve is as long as it doesn't intersect

itself the shoelace formula will always give the correct area. Here's an animated

complicated example in which I dynamically update what area the

shoelace formula has arrived at at the different points of the radius

changing sweeping direction.

Real mathematical magic, isn't it? It's also easy to see why reversing the

sweeping direction leads to negative area. Let's see.

Sweeping in the counterclockwise direction we first come across (a,b) and

record it, followed by (c,d). When we sweep clockwise

the order in which we come across (a,b) and (c,d) is reversed and this leads to

these changes in the formulas. And the last swap obviously leads to the number

turning into it's negative. And that's really it. Now you know how the shoelace

formula does what it does. In these videos we keep encountering really fancy

curves like this cardioid in a coffee cup in the "Mandelbrot and times tables"

video or this deltoid rolling curve whose area actually already played a

quite important role in the video on the Kakeya needle problem. At first glance it

looks like we won't be able to use the shoelace formula to calculate the area

of one of these curves because they are not made up from line segments. Well you

can definitely approximate the area by calculating the area of a straight line

approximation like this, with those blue points on the curve. And by increasing

the number of points we can get as close to the true area as we wish. In fact, by

taking this process to the limit in the usual calculus way, we can turn the

shoelace formula into a famous integral formula for calculating the exact area

enclosed by complicated curves like the deltoid. Here's how you do this. I've

tried to make sure that even if you've never studied calculus you'll be able to

get something out of this. Well we'll see, fingers crossed :) A curve like this is

often given in parametric form. For example this is a parametrizations of

this deltoid. Here x(t) and y(t) are the coordinates of a moving point that

traces the curve as the parameter t changes from, in this case 0 to 2 pi. Let's

have a look. So here's the position of the point at t=0.

And once it gets going the slider up there tells you what t we are

up to. Right now we'll translate all this into the language of calculus. Let's stop

the point somewhere along its journey. A little bit further along we find a

second point. A tiny, tiny little bit further on is usually expressed in terms

of infinitesimal displacements in x and y. It's a bit lazy to do it this way but

mathematicians are a bit lazy and love doing this because it captures the

intuition perfectly and in the end can be justified in a rigorous way. Anyway

just add dx and dy to the coordinates of our first point to get the coordinates

of our second point. Now, of course, these displacements are not independent of

each other. The connection is most easily established in terms of the derivatives of

the coordinate functions. So the derivative of the x coordinate with

respect to the parameter t is dx/dt which I write at x'(t) and

similarly for the y coordinate function. Solving for dx and dy gives this and this

then links both dx and dy to an increment dt of the parameter t that's

changing, right?

Now we substitute like this and now we're ready to calculate

the area of our infinitesimal triangle as before.

1/2 times a cross. And this evaluates to this expression here. And this we can

write in a slightly more compact form like that. Okay now what we have to do is

to add all these infinitely many infinitesimal areas and as usual in

calculus this is done with one of those magical integrals. The little circle

twirling in the counterclockwise direction

says that we're supposed to integrate around the curve exactly once in the

counterclockwise direction. Well let's see: for our deltoid we have

this parameterization here. We've already seen that a full trace is accomplished by

having t run from 0 to 2 pi. This means that in this

special case our integral can be written like this. Now evaluating and simplifying

the expression in the brackets gives this integral here, which can be broken

up into two parts. Maths students won't be surprised that the trig(onometric) integral on the

right evaluates to 0 which then means that the area where after is equal to this

baby integral which of course is equal to 2 pi. Now the little rolling circle

that is used to produce our deltoid is of radius 1 and is therefore of area pi.

This means that the area of the deltoid is exactly double the area of the

rolling circle. Neat isn't it? Okay, up for a couple of

challenges? Then explain in the comments what the number stands for that the

shoeless formula or the integral formula produce in the case of self intersecting

curves like these here. Another thing worth pondering is how the argument for our

triangle formula has to be adapted to account for the blue points ending up in

different quadrants, for example, like this. And that's it for today. I hope you

enjoyed this video and as usual let me know how well these explanations worked

for you. Actually since I mentioned the Kakeya video and fish,

I did end up turning my Kakeya fish into a t-shirt. What do you think? Well

and that's really it for today.