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Area of a circle intuition

Video transcript

- [Voiceover] What I'd like to do in this video is make an informal argument for why the formula for the area of a circle is Pi r squared. And we're gonna start just with the most traditional definition of the number Pi and that's that and I'll just do it here in a corner some place that Pi is equal to the ratio of the circumference and the diameter of the circle or the ratio of the circumference to the diameter of a circle or, of course, we could write this as ratio of the circumference to, instead of the diameter, I could write two times the radius or I can multiply both sides times two times the radius and we get our traditional formula for the circumference, for the circumference of a circle, but once again, this literally just comes straight out of the definition for the number Pi. The number Pi is defined as the ratio between the circumference and the diameter, so you just multiply both sides out of it times the diameter, you get the circumference is equal to Pi times the diameter. Now, we have the circumference formula right here, once again, this comes out of the definition of Pi, but from this, I'd like to at least get an intuitive feel for why the area formula is given by Pi r squared and to think about that, we're going to approximate the area of polygons, the areas of polygons, that are inscribed in a circle. So over here, I have this, what's this? There's a five-sided polygon right over here and its area is going to be equal to, so the area of this polygon, it's going to be five times the area of each of those triangles and the area of each of those triangles, the height is a, the base is b, so it's going to be base times height or height times base times 1/2. So this, once again, five times ab over two, it's not that good of an approximation. This would be the area of just the inscribed polygon, so we're definitely underestimating the area of the entire circle or leaving out all of these little, these little chunks outside of the polygon, but still inside of the circle. But as we add more size to the polygon, we see that we're leaving out less. We see, when we have now, this is a one, two, three, four, five, six, seven-sided polygon, we're leaving a little bit less. We're underestimating still, but we're underestimating by less. This area that we're giving up isn't as large as this area right over here. So in this approximation, we have, what did I say? Seven triangles? One, two, three, four, five, six, seven triangles. And the area of each of those triangles is once again ab over two. Now, a and b here are different than a and b over here, and notice what's happening. As we increase, as we increase the number of triangles, not only is it approximating the area of the circle better, but a is getting longer. And you can see, you could imagine, as we increase many, many, many more triangles, a is going to approach r. Now another thing to think about is what is seven, what is seven times b approaching? So we're saying that a is approaching r as we add more of these sides of the polygon, as we add more triangles, now what is the number of triangles times the base of the triangle, what is that approaching? Well this is going to approach the perimeter of the, or this is going to be the perimeter of the polygon, so seven times b is that plus, let me actually, let me draw this, it's that plus that plus that, I think you get the point, plus that, plus that, plus that, plus that. So once again, seven, let me write this at seven, times b, that is the perimeter of the polygon, perimeter, perimeter of the polygon. So think about what's happening. As we have more and more sides of the polygon, our a, our height of each of our triangles, is going to approach our radius, is going to approach the radius. It's going to get the height of each of the triangles. It's gonna get longer and longer, it's gonna approach the radius as we have, as we approach an infinite number of triangles, and then the number of polygons we have, not the number of polygons, the number of sides we have times the bases, that's going to be the perimeter of the polygon and as we add more and more sides, as we add more and more sides, the perimeter of the polygon is going to approach, is going to approach, is going to approach the circumference of the circle. I'll write it out, circumference, circumference, and you'll see that even more clearly right over here. So, once again, how many sides do I have here? I have one, two, three, four, five, six, seven, eight, nine, 10 sides, so this, I can write the perimeter of the polygon as 10 times b and then if I multiply that times a over two, if I multiply that times, we'll use another color. a over, let me just write it like this, times a over two, I'm once again approximating the area of the circle 'cause a times b over two, that's the area of each of these triangles and then I have 10 of these triangles, but now let's think about this more generally. Let's think about it if I were to have n, if I were to have an n-sided polygon, so I have n-sided polygon, then I'd be approximating the area as n times b, n times b, we see this right over here. When n is equal to 10, you have 10 times b. So it's n times b times a over two. Times a over two. I just wrote, this isn't something mysterious. The base times the height divided by two, this right over here, that's the area of each triangle and then I'm gonna have n of these triangles, so this is our approximation for the area, so let me write this, the area is going to be approximately that right over there. It's going to be n, the number of triangles I have, times the area of each triangle. Now what's going to happen, as n approaches infinity, as I approach having an infinite sided polygon, as I have an infinite number of triangles, so let's just think this through a little bit. 'cause this is where it gets interesting. Now this is the informal argument. To do this better, I'd have to dig out a little bit of calculus, but this gives you the essence. So let's just think about what happens as n approaches infinity. So, as n approaches infinity, we've already said, as we have more and more sides and we have more and more triangles, a approaches r, so let's write that down. So, a is going to approach r, the height of the triangles is going to approach the radius and what else is going to happen? Well, n times b, the perimeter of the polygon, the perimeter of the polygon is going to approach the circumference. So, a is going to approach r and n times b is going to approach the circumference, is going to approach the circumference, or another way of thinking about it, if it's approaching the circumference, we could say that n times b is going to approach two Pi times the radius 'cause that's what the circumference is going to be equal to. So, if a is approaching the radius and nb is approaching two Pi r, well then, what is the entire, what is the area of, what is the area of our polygon or what is the area of our polygon going to, or, the area of our circle going to be? Well, it is going to approach, it is going, or I should say, the area of our polygon is going to approach, nb is going to approach two Pi r, nb is going to approach two Pi r. Instead of nb, I'm writing two Pi r there. a is going to approach r. a is going to approach r, and then I'm dividing it, and then I am dividing it by two. So as n approaches infinity, as we have infinite number of sides of our polygon, an infinite number of triangles, the area of our polygon will approach this, which is equal to what? Well you have two divided two and then Pi r times r is equal to Pi r squared. So as we approach having infinite number of triangles and infinite, I wanna keep that there, infinite number of sides, we see that we approach the area of the circle and as we approach the area of the circle, we are approaching Pi r squared. So hopefully this gives you an intuitive sense why this right over here is the formula for the area of a circle. You could think about it as the area of an infinite sided polygon that is inscribed in the circle, which will be equal to the area of the circle.