Math Detours · Grade 6 · Around the corner · 22

Why a pizza is πr²

You have probably been handed \(A = \pi r^2\) as a fact. Tonight you see where it comes from, using nothing but a pizza cutter, and then use it on a rope wrapped around the whole planet.

Name ____________________Date ____________

Tonight's idea

Start with the circumference. Wrap a piece of string around any can, measure it, and divide by the width of the can. You always get about 3.14. That ratio is \(\pi\), and it is the same for every circle, so circumference \(= \pi d = 2\pi r\). The Greeks knew this; what took longer was the area.

Here is the argument. Cut a pizza into many thin slices, say 16. Lay them in a row, points alternating up and down, so they interlock. The shape is nearly a rectangle. Its height is the radius \(r\) (the length of a slice). Its width is half the crust, because half the slices have their crust on top and half on the bottom: \(\tfrac{1}{2} \times 2\pi r = \pi r\).

→ height ≈ r width ≈ half the crust = πr
Sixteen slices, points alternating. More slices make a straighter rectangle.

Area of a rectangle is width times height: \(\pi r \times r = \pi r^2\). With 16 slices the edges are a little bumpy. With 1,000 slices they are nearly straight, and the bumps can be made as small as you like. That "as small as you like" is the idea behind calculus, which you will meet in a few years.

You can also find \(\pi\) by throwing random points at a square and counting how many land inside a circle. Try the darts demo.

Watch one: the rope around the Earth

A rope lies snugly around the equator, about 40,000 km long. You add just 1 metre and spread the slack evenly, so the rope floats at the same height all the way round. How high? Enough for an ant? A cat?

Old radius \(r\), new radius \(r + h\). The new circumference is \(2\pi(r + h) = 2\pi r + 2\pi h\), which is 1 metre longer: \(2\pi h = 1\), so \(h = 1 / 2\pi \approx 0.16\) m, about 16 cm. A cat fits. Notice \(r\) vanished: the answer is the same for a rope around a basketball or around the Sun.

Your turn

  1. A circular table has radius 60 cm. Find its circumference and its area. (Use \(\pi \approx 3.14\).)
  2. A string around a can measures 26 cm. What is the can's diameter?
  3. A 14-inch pizza costs $18 and a 10-inch costs $10. Which is the better deal per square inch? First compute how many times bigger the 14-inch is.
  4. On a running track the lanes are 1.22 m wide. In a full lap, how much further does lane 2 run than lane 1, if both keep to the inside of their lane? Does your answer depend on the size of the track?
  5. A bicycle wheel has diameter 70 cm. How many times does it turn in 1 km?
  6. Explain the pizza-slice argument to someone in your own words, using a drawing with 8 slices. Why do more slices make the argument better?

Stretch: rings and Archimedes

A ring (the mathematical word is annulus) has outer radius 5 cm and inner radius 3 cm. Find its area. Then find the area of a 60° slice of a circle of radius 6 cm.

Archimedes, around 250 BC, trapped a circle between two 96-sided polygons, one just inside and one just outside, and proved that \(\pi\) is between \(3\tfrac{10}{71}\) and \(3\tfrac{1}{7}\). Turn both into decimals. How many decimal places of \(\pi\) did he pin down?

Answers

  1. Circumference \(2 \times 3.14 \times 60 \approx 377\) cm. Area \(3.14 \times 3600 \approx 11{,}300\) cm², about 1.13 m².
  2. \(26 / 3.14 \approx 8.3\) cm.
  3. 10-inch: area about 78.5 in², so \(1000 / 78.5 \approx 12.7\) cents per square inch. 14-inch: about 153.9 in², 11.7 cents per square inch. The 14-inch is the better deal: it is \((14/10)^2 = 1.96\) times the pizza for 1.8 times the price.
  4. Extra distance \(= 2\pi \times 1.22 \approx 7.67\) m per lap. It does not depend on the track's size at all, for the same reason as the rope around the Earth. That is why staggered starts on a 400 m track are about 7.7 m per lane.
  5. Circumference \(= 3.14 \times 0.7 \approx 2.2\) m, so \(1000 / 2.2 \approx 455\) turns.
  6. Key points: slices alternate up and down; height is a radius; width is half the crust because half the crusts are on top; more slices make the top and bottom flatter and the sides more vertical, so the shape approaches a true rectangle.

Stretch. Annulus: \(\pi(5^2 - 3^2) = 16\pi \approx 50.3\) cm². Sector: 60° is one sixth of a circle, so \(\tfrac{1}{6} \times \pi \times 36 = 6\pi \approx 18.8\) cm². Archimedes: \(3\tfrac{10}{71} \approx 3.1408\) and \(3\tfrac{1}{7} \approx 3.1429\). Both start 3.14, so he had two decimal places, and knew the third was between 0 and 2.

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