To the nearest square inch, what is the surface area of a balloon which is 20 inches in diameter?

Calculating the amount of material in a perfectly spherical balloon requires the classic geometric formula for the surface area of a sphere: 4 * pi * r^2. A common pitfall is to use the diameter directly, but the formula requires the radius (r), which is always half the diameter. For a balloon 20 inches across, the radius is 10 inches. Plugging this value into the formula gives us 4 * pi * (10^2), which simplifies to 4 * pi * 100, or 400 * pi. Using the value of pi (~3.14159), the result is approximately 1256.64 square inches, which rounds up to the final figure.

What’s fascinating about this formula is its elegant connection to another shape, a discovery made by the ancient Greek mathematician Archimedes. He proved that the surface area of a sphere is exactly the same as the lateral (side) area of a cylinder that would perfectly enclose it. For our 20-inch balloon, this "hat-box" cylinder would be 20 inches tall and also have a 10-inch radius. The area of its side would also calculate to 400 * pi, providing a brilliant visual proof for the formula. Archimedes was so proud of this discovery that he reportedly had a sphere and cylinder engraved on his tombstone.