The Circular Food
Drag the point \((a,a)\)
to move all four cuts.
Move the Cuts
Each pair of graphs shares the same value of \(a\). In the 2D section, drag the point on the food, drag a moving point on the area graph, or use the slider. Change the food too—the picture changes, but the mathematics does not.
Part 1 — Cutting a Circle
A pancake was where the question started, but the same geometry works for all kinds of circular foods.
Cut distance \(a\)
0.780
Circular food
Waffle
Area vs. Cut Position
Drag any moving point horizontally.
The food graph follows automatically.
Orange center
—
One blue side
—
One green corner
—
The curves compare one piece of each type.
The picture shades all four symmetric blue pieces
and all four symmetric green pieces.
\[
A_{\text{orange}}
+4A_{\text{blue}}
+4A_{\text{green}}
=
\pi R^2
\]
Part 2 — What If the Pancake Were a Ball?
The four 2D cuts become six planes. The sphere now contains a center cube, six face pieces, twelve edge pieces, and eight corner pieces.
Cut distance \(a\)
0.223
Sphere Decomposition
Each color button turns every
symmetric region of that color
on or off.
Volume vs. Cut Position
Drag any black point horizontally.
The 3D object changes with it.
Orange cube
—
One blue face
—
One green edge
—
One red corner
—
The picture and graph are showing
two related things.
The 3D picture displays all symmetric
pieces of each visible color:
1 orange, 6 blue, 12 green, and 8 red.
The graph measures the volume of
one representative piece
of each color.
\[
V_{\text{orange}}
+6V_{\text{blue}}
+12V_{\text{green}}
+8V_{\text{red}}
=
\frac{4}{3}\pi R^3
\]
\(1+6+12+8=27\) regions fill the sphere.
Where It Started
Start with four straight cuts through a circular pancake. Moving one parameter creates a whole family of cutting problems.
The food itself is not important. A waffle, pizza, cake, cookie, pie, cracker, or durian gives the same geometry.
What to Notice
The graph on the right is another way of seeing the geometry on the left.
When two curves cross, two differently shaped pieces have the same area or volume.