After Dinner, a Tiny Slice of Pi

One evening, after the dinner dishes had been cleared and our kitchen table had made its nightly transformation into a laboratory for mathematical exploration, I proclaimed my dad a true magician. How else could he have proved, before my very eyes, that the area of a circle was indeed πr²? As a 10-year-old, I was enraptured by Dad’s showmanship, which cleverly disguised a lesson in basic geometry.

By my third birthday, Dad–an electrical engineer–had instilled in me a fascination and familiarity with numbers. At eight, having grown bored with my schoolwork, I began sitting at the table and looking on as my father helped my older sister, Caroline, with calculus. In that kitchen I learned about algebra, geometry, and precalculus years before I would encounter them in school. A proud immigrant and doting father, Dad dreamed of his two daughters’ attending MIT. His dream came true, but our science careers were not nearly as long and storied as his has been. We paved our own paths after graduating from MIT: Caroline attended Harvard Business School, and I left organic chemistry for medical writing.

At the kitchen table on the night of the magic trick, Dad had asked, “Do you remember how to find the area of a circle?”

“Pi … something,” I stammered. Dad proceeded to use a compass to draw a large circle on a sheet of thick white paper. Next, he asked me to imagine that the circle was a pie with many slices. He looked on intently as I drew 20 or 30 “slices.” Then he produced a pair of scissors, and I cut out the triangular wedges. Absorbed in the task, I soon forgot about geometry. When I finished, I looked up expectantly.

Dad held up one of the slices–long and skinny, with a short curved crust. “What is the length of this slice in relation to the circle?” he asked.

“R!” I exclaimed, to Dad’s approving nod. “That length is the radius.”

“Okay, that was the warm-up,” he said. “Now for the challenge: if you add up the length of all the curved parts of the slices, what do they sum up to in relation to the circle?”

The curved parts made up what was originally the outside of the circle before I had cut it into pieces. “The circumference!” I responded triumphantly.

“And what is the circumference in relation to the radius?” he probed further.

“It’s pi times the diameter,” I recalled, “which is twice the radius. Two pi r. That’s easy. So what?”

“You grossly underestimate my mathe­matical abilities,” he said in a grandiose manner. “Assemble these slices into a rectangle.”

It was at this point that the simple exercise began to seem more like magic than math. I puzzled over the slices for some time before arranging them in an alternating pattern, one piece pointing up and the next piece pointing down. Finally, they wedged together perfectly, with the curved, scalloped edges forming the long sides of a rectangle. “Done!”

“Very good,” Dad said. “Now, what is the area of a rectangle?”

“Length times width,” I parroted impatiently. But what did the rectangle have to do with the area of the circle? Dad smiled and waited. And just as he expected, something clicked. This had originally been a circle, but it was now reshaped as a rectangle …

Suddenly, it all became clear: the width was r, the radius of the former circle. The length was half of the circumference, or πr. Pi times r times r was … “pi r squared!” I announced, mesmerized by my father’s “trick”–and my very first mathematical derivation.

Dad’s lessons–which were often embedded in discussions of music, finance, history, medicine, genetics, politics, and even sports–have had the power to transcend their subject matter. So it’s not surprising that when I encounter a particularly troubling decision, I still draw on his trusty mathematical advice. First, I list the givens, or what I know to be right and true. As is often the case in algebra, the act of writing down what I do know may be enough to uncover what uncertainties still remain. Next, I’m careful to label all the unknown variables: x, y, z. I draw parallels to connect past circumstances to situations I’m facing in the present. And I never forget to use Dad’s favorite power tool, dotted “auxiliary lines”–which, in life as in geometry, help me extrapolate beyond what’s in plain view, allowing a clearer picture of all the possibilities ahead.

Deborah Pan ‘03, who majored in chemistry and minored in writing, says her father dreamed of attending MIT while growing up in Kaohsiung, Taiwan. Deborah currently works as a medical editor and publications consultant for the pharmaceutical industry.