Geometry Learning Tool

Pizza Theorem Explorer

Move the cutting point away from the center, change the number of slices, and see when alternating slices share the pizza equally.

Person-1 takes the odd-numbered slices. Person-2 takes the even-numbered slices.

Pizza Theorem: If a circle is divided into 2n equal-angle sectors by n straight cuts through a common point, with n even and n at least 4, then the sums of the areas of alternating sectors are equal. Person-1 takes slices 1, 3, 5, 7... and Person-2 takes slices 2, 4, 6, 8...

Interactive pizza cuts

Choose 4, 8, 12, 16, or more slices. Every slice is numbered, and the Person-1 odd group and Person-2 even group update as you move the common cutting point.

Drag guide: Option 1: drag the blue cutting point inside the pizza. Option 2: use the sliders on the right, or below on phones, to move the same cutting point.
Slices8
Straight cuts4
Person-1 odd slices50.0%
Person-2 even slices50.0%
Balanced: difference 0.0% Each slice angle at the point: 45 degrees Cutting point: a little right and up
8 slices
-18 degrees
Right 28%
Up 16%
Angle gap45 degrees
Area difference0.0%
Person-1 slices1, 3, 5, 7
Person-2 slices2, 4, 6, 8
Slices = 2 x cutsEight slices come from four straight cuts. Twelve slices come from six straight cuts, and so on.
Person-1 total = Person-2 totalFor the theorem setups, starting with 8 slices, Person-1's odd slices and Person-2's even slices make two equal half-pizzas.

What is happening?

The point is away from the center, so some slices look larger than others. Person-1's odd slices and Person-2's even slices still balance because the cuts are evenly spaced.

Presets

Slice area notebook

Each small card shows one slice number, who takes it, and its approximate share of the whole pizza.

What students should remember

All cuts meet at one point

The cutting point can move away from the center, but every straight cut must pass through that same point.

The angles are evenly spaced

For 8 slices, each neighboring slice uses 45 degrees around the cutting point. For 16 slices, it uses 22.5 degrees.

Alternate slices are fair

Starting with 8 slices, Person-1 takes odd slices and Person-2 takes even slices. The totals stay equal even when the slice shapes look different.