Why Doubling a Baking Pan Quadruples Your Brownie Recipe

One day, while making brownies, we discovered the only available baking pan was twice as long as the one recommended in the recipe.

We first thought to double the ingredient quantities, but realized that filling the larger pan actually required four times the original amount.

When a pan has a fixed height, it can be treated as a two‑dimensional rectangle. Doubling the length doubles the area, and doubling the width doubles the area again, resulting in a total area increase of 2 × 2 = 4, so the cake becomes four times larger and the ingredients must be multiplied by four.

The same principle applies to any rectangle: scaling both length and width by a factor of 3 enlarges the area by 3² = 9, and scaling by a factor of 5 enlarges the area by 5² = 25.

More generally, when the side lengths are scaled by a factor of k, the area becomes k² times the original.

This rule is not limited to rectangles; it also holds for other two‑dimensional shapes such as trapezoids, triangles, circles, or any container you can pour batter into. Increasing linear dimensions always leads to a faster increase in area.

[US] Ben Orlin, translated by Tang Yanchi, “The Joy of Mathematics: A Book Full of Fun Illustrations”