Master Hypotenuse and Apprentice Sides

Master Hypotenuse went first. He set up his square and plumb line at the base of the first right-triangle stone. He measured the two legs of the right angle.

"This one is a right triangle," he said. "The legs are five hands wide and twelve hands tall. The hypotenuse — the side opposite the right angle — is the diagonal across the top. By the Pythagorean theorem, five squared plus twelve squared equals the hypotenuse squared. Twenty-five plus one hundred and forty-four is one hundred and sixty-nine. The square root of one hundred and sixty-nine is thirteen. So the hypotenuse is thirteen hands."

He wrote it down. Right triangle 1: legs 5 and 12, hypotenuse 13.

"And the area," he said. "For a right triangle, the area is one-half times leg times leg. One-half times five times twelve is thirty. So the area is thirty square hands."

He moved to the second right triangle. He measured its legs.

"Eight and fifteen. Eight squared is sixty-four. Fifteen squared is two hundred twenty-five. Sum is two hundred eighty-nine. Square root is seventeen. So the hypotenuse is seventeen, and the area is one-half times eight times fifteen, which is sixty."

He wrote it down. Right triangle 2: legs 8 and 15, hypotenuse 17. Area 60.

He came down to the ground. "Two right triangles, areas 30 and 60. The third one is the trouble. Its angles aren't ninety. I can climb up and measure the angle and then drop a perpendicular from the apex — but that means a ladder and a plumb line and probably an hour."

Apprentice Sides looked at him. "You don't need any of that."

Apprentice Sides climbed nimbly up to the gable's edge. She did not measure any angles. She did not drop any perpendiculars. She just measured the three SIDES of the third triangle with her tape measure.

"Side one: nine hands. Side two: ten hands. Side three: eleven hands."

She came back down. She wrote the three numbers on her wax-tablet.

"For any triangle whose three sides are a, b, c — even a non-right triangle — there is a formula for the area in terms of just the three sides. It's called Heron's formula. First you compute the semi-perimeter, which is half the sum of the three sides."

She wrote: s = (a + b + c) / 2 = (9 + 10 + 11) / 2 = 30 / 2 = 15.

"Then the area is the square root of s times s-minus-a times s-minus-b times s-minus-c."

She wrote: Area = √(15 × (15−9) × (15−10) × (15−11)) = √(15 × 6 × 5 × 4) = √(1800).

The square root of one thousand eight hundred was a little messy. She did the arithmetic carefully on the wax-tablet. About forty-two-and-a-half square hands.

She handed the wax-tablet to the Master Mason. "Area of the third triangle is about forty-two-point-four square hands. No ladder. No plumb line."

The Master Mason stared at the wax-tablet for a long moment.

"You just got an area from three side-lengths."

"Yes."

"Without any angle measurement."

"Yes."

He looked at Master Hypotenuse. "Do you know Heron's formula?"

"I know it exists," Master Hypotenuse said, slightly grudging. "I have always taught the right-triangle case because it's simpler. The Pythagorean theorem is one of the first things kids learn. Heron's formula is heavier; it has a square root nested inside a product, and most of the arithmetic gets messy."

"Right," Apprentice Sides said. "Yours is the cleaner tool for the easy case. Mine is the messier tool for any case."

The Master Mason wrote down the three areas. "Thirty. Sixty. Forty-two-point-four. I have everything I need."

He paused.

"There's a lesson in this for the apprentices," he said. "Which one of you tells it?"