# Understanding Why a Sphere's Surface Area Equals Its Curved Strip

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## Chapter 1: The Surface Area of a Sphere

Have you ever wondered why the surface area of a sphere is equivalent to the area of the cylindrical strip that encases it? Let’s explore this intriguing concept!

To illustrate, take a perfect sphere and encase it within a tube made of paper. The rectangular sheet required to form this tube has an area that matches the surface area of the sphere.

How can this be possible?

The area of the rectangle is easy to calculate. To surround a unit sphere, we need a height of 2 and a base length of 2π. Thus, the total area amounts to 4π. Simple enough!

However, the surface of the sphere features a complex curvature. The challenge lies in how we can map these curved sections onto flat ones.

Let’s divide the sphere, cylinder, and rectangle into numerous slices, each with a width denoted as dW.

### Section 1.1: Analyzing the Slices

Focusing closely on two individual slices, we can compare each slice from the sphere to its corresponding one from the cylinder.

The strips derived from the cylinder are uniform, whereas the upper strip from the sphere appears smaller than the equatorial strip.

For the areas to be equivalent, each circular strip must share the same area. Let’s zoom in on one quadrant of the sphere to analyze the radii and widths.

Here we observe the equatorial band with a width of dW and a radius of 1. Though this band doesn’t perfectly fit the sphere, we can reduce dW to an arbitrarily small size to achieve a closer fit.

Next, let’s examine the northern band.

The angle of the band’s latitude is denoted as θ, with the radius of its base being cos θ.

Finally, let’s take a closer look at a specific section of the previous figure.

Here, the band appears tilted, and its width is not simply dW, but rather the distance AB along the triangle.

#### Subsection 1.1.1: The Napkin Ring Paradox

An interesting observation is that any two spheres, hollowed out to share the same height, will possess the same volume. Here’s the reasoning behind this phenomenon:

As we move toward the north, the radius of the band decreases by a factor of cos θ. However, the distance between points A and B expands by the same factor. This distance defines the width of the band.

Each infinitesimal band retains the same infinitesimal area, irrespective of its position on the sphere. Each band corresponds precisely to one strip of the 2 by 2π rectangle.

Consequently, the unit sphere also has a surface area of 4π.

Bob’s your uncle!

## Chapter 2: Exploring the Concept Further

In this engaging video titled "But why is a sphere's surface area four times its shadow?", the creator delves deeper into the fascinating relationship between a sphere’s surface area and its shadow, providing a visual and intuitive explanation.

Another insightful video, "Surface Area of a Sphere", offers a detailed overview of how to calculate the surface area of a sphere and its geometric implications.