Friday, October 9, 2015
Unizor - Geometry3D - Cylinders - Area and Volume
Unizor - Creative Minds through Art of Mathematics - Math4Teens
Notes to a video lecture on http://www.unizor.com
Area and Volume of a Cylinder
For the definition of a cylinder and corresponding terminology please refer to topic "Elements of Solid Geometry" in this part of a course.
In short, a cylinder is formed by
(1) a generatrix (straight line) moving along a circular directrix perpendicularly to a plane where this circular directrix is located, thus making a side surface and
(2) two planes parallel to a plane that holds a directrix - top and bottom bases of a cylinder, they bound the cylinder from both ends..
Let's define two important parameters that fully characterize a cylinder.
(a) radius of a base circle, which we will refer to as radius of a cylinder,
(b) altitude or height of a cylinder (the distance between the top and bottom circular bases).
There are different approaches to defining an area of a cylinder. More rigorous approach involves full force of the theory of limits, but we would suggest here a different approach.
First of all, consider the side surface of a cylinder.
Since this side surface is formed by a straight line (generatrix) moving along a circular directrix always perpendicularly to the same plane where this directrix is located and, therefore, parallel to itself at different positions, it is intuitively obvious that, if we cut the side surface of a cylinder along one of these straight lines, we will be able to "flatten" it on a plane without stretching or squeezing, that is without any change to its area.
As a result of this transformation, we will obtain a rectangle with the width equal in length to a circumference of a circular base of a cylinder and the height equal to a height of a cylinder - both are known variables for any given cylinder.
Therefore, the area of a side surface of a cylinder is equal to the area of a rectangle and can be easily calculated.
The circumference of a circular base of a cylinder of a radius R equals to 2πR. If the height of a cylinder equals to H, the area of the side surface would be equal to 2πR·H.
The full area of a cylinder should include two areas of circular bases of radius R, each equal to πR².
That makes the full area of a cylinder of a radius R and height H equal to
2πR·H+2πR² = 2πR(R+H)
The situation with volume of a cylinder is less obvious and we will not be able to escape considerations based on the limit theory.
Let's inscribe into a circular base of a cylinder a regular N-sided polygon. Then construct a right prism with this polygon being a base and the height equal to a height of a cylinder. We obtain a prism inscribed into a cylinder.
Without rigorous proof, it is intuitively obvious that, as we increase the number of vertices N, the regular polygon inscribed into a circular base of a cylinder becomes closer and closer to a circle itself, and the prism, based on this polygon inscribed into a circular base of a cylinder, becomes closer and closer to a cylinder. So, the volume of a cylinder is a limit of the volumes of inscribed in this manner prisms as N→∞.
Since a volume of a prism is a product of an area of its base by height and, as N→∞, the area of the N-sided polygon inscribed into a circle of a radius R tends to the area of a circle itself, that is πR², while the height H remains constant, we conclude that the volume of a prism tends to πR²·H