If we assume side length of 5 (since it's a square)
The 4 triangles have base 4, height 2.5 = area 5
They also have 4 units of perimeter.
If you subtract 4*5 = 20 from the total area of 25, you find that the red areas total 5. Plus they have 1 unit of perimeter at each corner, for 4 total.
So yes, it is equally split 5 ways.
This technique also leads to a less rotationally symmetric division, but single pieces for each user:
"cut" it into 4 triangles, base 5 height 2.5 (centre point being the middle of each).
Put them in order so the "side" of the cake is along a line.
Mark the slices so that each has a base of 4, and cuts to the centre point. Fold it back up, make the cuts.
![[Image: cake2jx3.png]](http://img369.imageshack.us/img369/4117/cake2jx3.png)
![[Image: w528.png]](http://img369.imageshack.us/img369/cake2jx3.png/1/w528.png)
(I believe this is the same split used on the link given earlier)
Any split like this, done only with lines from the base to the point above them, with the length of the base for each person totalling 1/5 of the total, will be split equally. For example, cut each triangle into 5 pieces with base 1. Then give each person 4 of those pieces (of 20 total) for 1/5 of the cake each.
The 4 triangles have base 4, height 2.5 = area 5
They also have 4 units of perimeter.
If you subtract 4*5 = 20 from the total area of 25, you find that the red areas total 5. Plus they have 1 unit of perimeter at each corner, for 4 total.
So yes, it is equally split 5 ways.
This technique also leads to a less rotationally symmetric division, but single pieces for each user:
"cut" it into 4 triangles, base 5 height 2.5 (centre point being the middle of each).
Put them in order so the "side" of the cake is along a line.
Mark the slices so that each has a base of 4, and cuts to the centre point. Fold it back up, make the cuts.
![[Image: w528.png]](http://g.imageshack.us/img369/cake2jx3.png/1/)
(I believe this is the same split used on the link given earlier)
Any split like this, done only with lines from the base to the point above them, with the length of the base for each person totalling 1/5 of the total, will be split equally. For example, cut each triangle into 5 pieces with base 1. Then give each person 4 of those pieces (of 20 total) for 1/5 of the cake each.