Volume of a Cone, Part I
After I wrote this post, Vlorbik semi-challenged me to explain, sans calculus, how to find the formula for the volume of a cone.
Well, after what was likely about two miles worth of pacing--and with the help of a certain Greek who got to this about 1,700 years before I did--I think I came up with a simple (though windy) way of explaining how to find the formula.
I'll try to tackle this explanation in several parts, the first of which involves—as is the case with any understanding in mathematics—getting the "right" perspective.
For example, one can think about the volume of a cylinder as the movement of the area of its base (&pi × r2) a certain distance (h), giving us a volume of &pir2h for the cylinder.
Another way to use the movement of area to generate a cylinder is to rotate the area of a rectangle about one of the sides of the rectangle. The image on the right shows how this might be done.
Coming up with a volume formula based on this movement is a bit trickier. At first glance, one might be tempted to say that the rectangle's area (rh) moves a distance of 2&pir, which is the circumference, or distance around, the circular base. So we can simply multiply rh by 2&pir.
But this is not correct, because—and this is important—not every point of the area of the rectangle moves a distance of 2&pir.
Take a look at the diagram below.
The red points (all those points along the red line segment) will certainly move a distance of 2&pir, the circumference of the base. But the blue points are only halfway to the edge of the rectangle. As each of these points moves, they will carve out circles each with a radius of only 1/2r, meaning that each point on the blue line will move only 2&pi(1/2r), or &pir. And the green points are three fourths of the way to the edge of the rectangle, so they will move only 2&pi(3/4r), or 3/2&pir.
What to do, what to do?
Well, as it turns out, an ingenious method for dissecting this problem comes not from knowledge of geometry, but from knowledge of elementary data analysis. And that's where we'll pick it up in the next post.
Volume of a Cone: Part I | Part II | Part III | Part IV
Well, after what was likely about two miles worth of pacing--and with the help of a certain Greek who got to this about 1,700 years before I did--I think I came up with a simple (though windy) way of explaining how to find the formula.
I'll try to tackle this explanation in several parts, the first of which involves—as is the case with any understanding in mathematics—getting the "right" perspective.
Viva La Rotación!
A productive way to think about volume (in order to understand volume formulas) is to think about the movement of two-dimensional areas.For example, one can think about the volume of a cylinder as the movement of the area of its base (&pi × r2) a certain distance (h), giving us a volume of &pir2h for the cylinder.
Another way to use the movement of area to generate a cylinder is to rotate the area of a rectangle about one of the sides of the rectangle. The image on the right shows how this might be done.Coming up with a volume formula based on this movement is a bit trickier. At first glance, one might be tempted to say that the rectangle's area (rh) moves a distance of 2&pir, which is the circumference, or distance around, the circular base. So we can simply multiply rh by 2&pir.
But this is not correct, because—and this is important—not every point of the area of the rectangle moves a distance of 2&pir.
Take a look at the diagram below.
The red points (all those points along the red line segment) will certainly move a distance of 2&pir, the circumference of the base. But the blue points are only halfway to the edge of the rectangle. As each of these points moves, they will carve out circles each with a radius of only 1/2r, meaning that each point on the blue line will move only 2&pi(1/2r), or &pir. And the green points are three fourths of the way to the edge of the rectangle, so they will move only 2&pi(3/4r), or 3/2&pir.What to do, what to do?
Well, as it turns out, an ingenious method for dissecting this problem comes not from knowledge of geometry, but from knowledge of elementary data analysis. And that's where we'll pick it up in the next post.
Volume of a Cone: Part I | Part II | Part III | Part IV
Labels: mathematics