Unwrapping cylinder into a rectangle

From: 3blue1brown

A key step in understanding the surface area of a sphere involves the geometric principle of unwrapping a cylinder into a rectangle [00:00:56]. This technique helps to visualize and calculate the area of the cylinder’s curved surface.

Derivation of Cylinder’s Area

For a cylinder that encloses a sphere, if its circular caps are disregarded [00:00:13], its curved surface can be conceptually unwrapped into a flat, two-dimensional rectangle [00:00:56].

The dimensions of this resulting rectangle are defined by the original cylinder:

• One side length of the rectangle corresponds to the circumference of the cylinder’s base, which is 2πr (where r is the radius) [00:00:58].
• The other side length of the rectangle corresponds to the height of the cylinder [00:01:01]. In the specific case of a cylinder enclosing a sphere, this height is equal to the diameter of the sphere, or 2r [00:01:01].

Multiplying these two dimensions (2πr and 2r) gives the area of the rectangle: (2πr) * (2r) = 4πr² [00:01:05].

Connection to Sphere Surface Area

This method is significant because, according to Archimedes’ beautiful geometric reasoning, the surface area of a sphere is precisely equal to the area of the curved surface of a cylinder that perfectly encloses it [00:00:13], [00:00:51]. This connection is established through a projection of the sphere onto the cylinder, where small patches on the sphere cast shadows of identical area on the cylinder [00:00:27], [00:00:41].

Therefore, by unwrapping the cylinder, the formula for the surface area of a sphere, 4πr², is derived [00:01:05].

Comparison to Circle Area

The concept of unwrapping is also used to demonstrate the area of a circle. A circle can be unwrapped into a triangle where its height is the radius (r) and its base is the circumference (2πr) [00:01:12], [00:01:16]. Interestingly, four of these “unwrapped circles” fit perfectly into the unwrapped cylindrical shape [00:01:16].