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What is the area of a pizza?

By David Shaw, 14 March 2013

Pizzas come in a range of sizes. If you’ve ever wondered whether to get one big or two small pizzas, this experiment might be able to help!

Safety: When dealing with food, use clean hands and clean equipment!

You will need

• At least one pizza
• Knife
• Ruler or tape measure

What to do

1. Before you start, make sure the pizza is cool enough to touch – we don’t want you getting burnt!
2. Carefully cut the pizza into four pieces in the normal way (wedges which all meet in the middle). Try to make all the pieces the same size and shape.
3. Take two of the pieces and place them with the cut sides next to each other, but put the pointy end of one next to the crust on the other.
4. Add another piece to these two, keeping the pointy end of one piece matching the crust of the adjacent piece. Then add the final piece according to this same pattern.
5. Your pizza, which was a circle, should now be a bit more like a rectangle. Measure its width, and its length. Multiply the width by the length to calculate the area of your rectangular pizza. Is your pizza a perfect rectangle? Is this calculation the exact area of your pizza?
6. Carefully cut each piece in half, and then re-order the pieces so they follow the same pattern as before. Measure the pizza again and calculate its area. Is this a better estimate for the area of the pizza?
7. Cut the pieces in half yet again and reorder them. Does the pizza look more rectangular than it did before? Measure the pizza yet again and calculate its area. What do you think the actual area of your pizza is?
   

   Carefully cut the slices in half and rearrange them again.

8. If you have several different sized pizzas, you can follow these steps again with each pizza, then compare the areas. How much bigger is the biggest pizza compared with the smallest one? Are you surprised?

What’s happening?

Finding the area of a rectangle is a lot easier than most other shapes. All you need to do is multiply the length and the width. By cutting the pizza and rearranging it, you change it from being circular to being (almost) a rectangle. When you cut something up and rearrange the bits, you’ll end up with the same area at the beginning and end. So, measuring the area of a circle becomes as easy as measuring the area of a rectangle.

The rectangle that you made was not completely rectangular. The sides are made of crust, and the crust is curved. However, if you keep cutting the pizza and rearranging it, it will get closer and closer to a rectangle. If you could keep cutting and rearranging forever, then the pizza would make a perfect rectangle.

This idea of breaking something down into bits and rearranging them is very important in mathematics, and not just in geometry.

Applications

One nice thing about this activity is that once you’ve seen how it works, you don’t have to cut up every circle to measure the area. All you have to remember is that every circle can be turned into a rectangle in this way.

The short side of your rectangle is going to be the distance along one side of a piece of pizza. This distance is the same as the distance from the centre of your pizza to the edge.

The long side is made up of segments of crust. All of the crust runs along the two long edges, and the whole of those long edges are made from crust, so the length of the long sides is equal to half the length of the crust.

So the area of a pizza is the length of the crust (circumference) divided by two times the distance from the edge to the middle (radius). Mathematicians use a special number called pi to calculate the circumference of a circle. The circumference is two times pi times the radius (2πr). This means the area of a circle is pi times the radius, times the radius. This is more commonly written as pi r squared (πr²).

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