Can you cut a cake fairly to solve this middle school math problem?

[Whateverist]

Just as the area of the square representing the base of the cake alone must be divided into five equal pieces so long as the cuts are made from the center to five equally spaced points along the perimeter, so must the larger square consisting of the cake plus frosting.  Since the portions contained inside the square representing the unfrosted cake's base are equal, if you subtract them from each piece of the larger square representing the cake plus frosting, the remaining area (representing the base area of the frosted region alone) must also be equal.  Therefore the frosting is also divided equally in this solution, including the fours corner regions.

To see this I made a drawing to show directly that the base area of the frosted regions would still be equal when the frosting is applied to a thickness half the side length of the cake alone.   A larger scale version of the photo and an explanation of the drawing are below.



Larger scale photo:

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Explanation of the drawing:

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Okay so I drew a square cake 40 units on a side.  (This grid paper is ruled with a darker line every 8th line, or 8 lines to the inch.)  So the perimeter of the cake is 160 units which divides by 5 to make 32 units of perimeter per piece of cake.

I then drew very dimensional frosting, 20 units deep all the way around the cake.  The resulting cake has an 80 units by 80 square base. 

Next I drew cuts from the center to five points along the perimeter of the cake, each one 32 units from its neighbor all the way around the cake.  The cuts continue all the way through the cake and frosting.

The total area of the portion of the drawing representing the frosting, has a total area of 4800 sq. units, consisting of 4 squares at the corners of 20 by 20 and four rectangles between them of 20 by 40.  If the frosting is to be divided evenly each piece must contain 4800/5 = 960 sq. units of this base area.  

To find the base area of each of the five pieces I partitioned each one into rectangles and triangles and summed all its parts.  The resulting area worked out to be the 960 sq units required.

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Green, blue and purple gets a whole corner each. Red and yellow have to share one.

Notice that though the yellow and red pieces get only half a corner of frosting, each one gets portion of frosting that extends into the area directly above its neighbor.