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### Dividing a Square Cake into Five Equal Pieces

```
Date: 07/28/2001 at 10:48:48
From: Pam English
Subject: Dividing regions of a square

How can you divide a square-topped cake that is a rectangular solid
and is frosted on all faces into five pieces so that everyone receives
the same amount of cake and icing? All cuts must be perpendicular to
the surface.

I can not figure this out. All I can do is divide the cake four ways
solve this problem.
```

```
Date: 07/28/2001 at 22:20:31
From: Doctor Peterson
Subject: Re: Dividing regions of a square

Hi, Pam.

This problem is easier to solve than you would think. I suggest you
start by thinking about wedge-shape pieces like this:

s
+---------------+
|               |
|               |
|               |
|       +       |
|      / \      |
|     /   \     |
|    /     \    |
+---+-------+---+
a

Then you can extend it to wedges that wrap around a corner like this:

s
+---------------+
|               |
|               |
|               |
|       +       |
|      /    \   |
|     /         +
|    /          |b
+---+-----------+
a

You can find this area by drawing the line from the center to the
corner to divide it into two triangles, and finding the area of each
triangle.

Once you've done this, I think you will have a good idea what to do.

For a more detailed answer in our Dr. Math archives, see:

Cutting a Square into Five Equal Pieces
http://mathforum.org/dr.math/problems/shinichi.7.12.99.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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