Monday, November 2, 2015
Unizor - Geometry3D - Cones - 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 Cone
Let's define two important parameters that fully characterize a cone.
(a) radius of a base circle, which we will refer to as radius of a cone and denote as R,
(b) altitude or height of a cone (the distance from an apex to a bottom circular base), which we denote as H.
Note that the distance from an apex to any point on a circular base it is connected with by a straight line on a side conical surface is constant and, according to Pythagorean Theorem, is equal to
L = √R²+H²
There are different approaches to defining an area of a cone. 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 cone.
Since this side surface is formed by a straight lines connecting points of a circular directrix with an apex, it is intuitively obvious that, if we cut the side surface of a cone 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 circular sector with radius equal to a distance between an apex and each point on a circular base of a cone that we have calculated above as
L = √R²+H².
Another characteristic of this sector, that we will use to determine its area, is the length of an arc of this sector. Obviously, it is equal to a circumference of a base of a cone, that is
C = 2πR.
Therefore, the area of a side surface of a cone is equal to the area of a circular sector with a radius
L = √R²+H²
and an arc length
C = 2πR.
The sector's area is a part of an area of a circle of the same radius L. The ratio between the sector's area and the area of a circle, which this sector is a part of, equals to the ratio between their arcs.
We know the arc of a sector, it is equal to 2πR.
We also know the arc (that is, circumference) of a circle, which our sector is a part of, it is equal to 2πL.
And we know this circle's area, it is equal to πL².
We can determine the side area Sside of a cone from the following ratio:
Sside /πL² = 2πR /2πL
The solution to this is
Sside = πR·L = πR√R²+H²
To determine a full area of a cone, we have to add an area of its base Sbase=πR².
The final result for a full area of a cone Sfull:
Sfull = Sside + Sbase =
= πR√R²+H² + πR² =
The situation with volume of a cone is similar to that of a volume of a cylinder, and we will not be able to escape considerations based on the limit theory.
Let's inscribe into a circular base of a cone a regular N-sided polygon. Then construct a pyramid with this polygon being a base and the same apex as that of a cone. We obtain a pyramid inscribed into a cone.
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 cone becomes closer and closer to a circle itself, and the pyramid, based on this polygon inscribed into a circular base of a cone, becomes closer and closer to a cone. So, the volume of a cone is a limit of the volumes of inscribed in this manner pyramids as N→∞.
Since a volume of a pyramid is one third of 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 pyramid tends to
V = πR²·H /3