The volume enclosed by a sphere of radius r is (4/3)πr^3. The surface area of the same sphere is 4πr^2. You may already have noticed that the volume is exactly (1/3)r times the surface area. Explain why this relationship should be expected. One way is to consider a billion-faceted polyhedron that is circumscribed about a sphere of radius r; how are its volume and surface area related?
The volume of a pyramid is 1/3 Bh
Consider the many-faceted polyhedron as a collection of pyramids.
Each small patch -- of area, say, B, of the surface of the sphere is the base of a pyramid of height r, giving it a volume of 1/3 Br.
Adding up all the patches, you get the surface of the sphere, multiplied by 1/3 r, giving the volume of the sphere.
To understand why the relationship between the volume and surface area of a sphere can be expected, let's consider the concept of surface-to-volume ratio.
The surface area of an object represents the amount of material or space that can interact with its surroundings. In the case of a sphere, the surface area represents the total area of all its sides or faces.
On the other hand, the volume of an object represents the amount of space it occupies or the amount of material it contains.
Now, let's consider the billion-faceted polyhedron that is circumscribed about a sphere of radius r. The polyhedron will have a larger surface area due to the presence of numerous faces, while the sphere will have a smaller surface area as it has only a single continuous curved surface.
As the number of facets or faces on the polyhedron increases and tends to infinity, the polyhedron starts to resemble a sphere more closely. In the limit, when the number of faces becomes infinitely large, the polyhedron will converge to a perfect sphere.
When considering the relationship between the volume and surface area of this polyhedron, we can see that as the number of faces increases, the volume of the polyhedron will approach the volume of the sphere, and the surface area of the polyhedron will approach the surface area of the sphere as well.
Since the volume and surface area of the sphere are both finite, this relationship suggests that the volume of the sphere is proportional to the surface area. In other words, the volume could be expressed as a constant multiplier times the surface area.
By comparing the formulas for the volume and surface area of a sphere, we can see that the volume is indeed exactly (1/3)r times the surface area:
Volume = (4/3)πr^3
Surface Area = 4πr^2
Now, we can express the volume in terms of the surface area:
Volume = (4/3)r * Surface Area
Therefore, it is expected that the volume of a sphere is directly related to the surface area by a factor of (1/3)r.
To understand why the relationship between the volume and surface area of a sphere holds, let's consider a billion-faceted polyhedron that is circumscribed around a sphere of radius r.
When this polyhedron is sufficiently large and has a large number of facets, it approaches the shape of a sphere. The more facets it has, the closer it becomes to a precise sphere. Additionally, because the polyhedron is circumscribed around the sphere, its vertices (corners) touch the sphere's surface, and its edges connect the vertices without intersecting the sphere.
Now let's examine how the volume and surface area of this polyhedron are related. First, consider the surface area. As the number of facets increases, the polyhedron becomes more finely divided, resulting in smaller and smaller faces that approximate the curvature of the sphere. In the limit as the number of faces approaches infinity, each individual face becomes infinitesimally small and almost perfectly flat compared to the sphere's overall curvature. Consequently, the total surface area of the polyhedron approximates the surface area of the sphere.
Next, let's look at the volume. The volume of the polyhedron can be calculated by summing the volumes of its individual facets. As the number of facets increases, the polyhedron becomes more and more like a solid shape. In the limit as the number of faces approaches infinity, the polyhedron becomes a perfect sphere, and its volume approximates the volume of the sphere.
Now, let's explore the ratio between the volume and surface area. As the number of facets increases and the polyhedron approaches a perfect sphere, the ratio of its volume to surface area remains constant. This is because the ratio depends solely on the shape itself, irrespective of the size or number of facets.
This ratio, which we noticed earlier as (1/3)r, holds true for the polyhedron approximating the sphere. Since the polyhedron approaches a sphere as the number of facets increases, we can conclude that the same ratio, (1/3)r, applies to the sphere as well.
So, in essence, the reason why the volume of a sphere is (1/3) times its surface area is due to the nature of its perfectly symmetrical, smoothly curved shape. The billion-faceted polyhedron helps us visualize this relationship by approximating the sphere and showing how its volume and surface area are connected.