Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the seventh episode of Proven Yet Hard to Believe, a series where we uncover the fascinating stories and directly explore the core ideas behind mathematical discoveries that have stood the test of time. This time, we’re once again diving into the genius of Archimedes—an ancient thinker whose groundbreaking ideas continue to amaze and inspire us even today.
Are you ready to journey into his brilliant mind?
Let’s get started!
Archimedes discovered the volume of a sphere and used it as the basis to calculate its surface area.
Let’s follow his thought process step by step.
First, he imagined dividing the surface of a sphere into tiny shapes. Then, he visualized the sphere as being composed of small polyhedral shapes that could be seen as pyramids. Specifically, these were shapes resembling pyramids, with their bases on the sphere’s surface and their tips meeting at the sphere’s center.
Using the formula for the volume of a pyramid—one-thirds times base area times height—Archimedes began summing up these tiny volumes. Since the height of each cone is equal to the sphere’s radius, he factored it out and rearranged the equation like this….
The result? The total volume is equal to that of the sphere itself. From there, the surface area of the sphere could be derived.
However, Archimedes considered this reasoning more of an idea than a rigorous proof. In his own words, it was simply a way to gain “some prior knowledge” about the questions at hand.
He went on to develop a precise geometric proof using the method of exhaustion, an early form of integration. Here’s the core idea of that proof.
Archimedes imagined stacking frustums (truncated cones) both inside and outside the sphere. This approach allowed him to calculate not just the sphere’s volume but also its surface area. To better explain, let’s take a cross-section of the sphere and focus on one small part.
Instead of thinking of the arc as a curve, Archimedes treated it as a straight line segment of length s. This segment is very short and lies tangentially along the sphere. He then wrapped a thread tightly around this segment, creating a dense covering without overlaps.
The total length of the thread was equal to the lateral surface area of a thin frustum. Since the thread slanted slightly, the circumferences of the circles in the stack varied. However, by pairing these circles, Archimedes found that their combined circumference was equal to that of two circles with a radius b, where b represents the distance measured from the vertical line passing through the center of the sphere.
Using the similarity of shapes, Archimedes calculated the total length of the thread. Remarkably, the surface area of this section turned out to be identical to that of a cylinder with the same height.
In conclusion, the sphere’s total surface area equals the lateral surface area of a cylinder that encloses the sphere. This insight remains one of the most elegant mathematical revelations in history.
And there you have it—Archimedes’ brilliant way of calculating the surface area of a sphere.
Isn’t it incredible how his ideas from over two millennia ago still inspire us today? Thank you for joining me on this journey into mathematical history. Until next time, keep questioning, keep exploring, and stay curious.
See you soon!