A cylindrical soup can four inches in diameter and four inches tall is completely surrounded by a paper label. What is the area of that label? (to the nearest whole number)

To solve this classic geometry puzzle, the key is to visualize peeling the label off the can and laying it flat. When you do this, the curved label unrolls into a simple rectangle. The height of this rectangle is easy—it’s the same as the height of the can, which is four inches. The challenge is figuring out the rectangle's width.

The width of the flattened label is equal to the distance around the circular top of the can, a measurement known as the circumference. The formula for finding the circumference is pi (approximately 3.14159) multiplied by the diameter. For this can, that's pi times four inches, which equals about 12.57 inches. Now you have the dimensions of your rectangular label: a height of 4 inches and a width of about 12.57 inches.

To find the area, you simply multiply the height by the width. So, 4 inches multiplied by 12.57 inches gives you 50.28 square inches. When rounded to the nearest whole number, the area is 50 square inches. This calculation, known formally as finding the lateral surface area of a cylinder, is essential in manufacturing and packaging to determine the exact amount of material needed for labels, wraps, and pipes without costly waste.