**Complete
Solution:**
There are many ways to approach this problem.
Assuming that all pieces of the cake have the same height,
the size (or volume) of each piece depends on the area
of its top. **Since icing is on
both the top and the sides, however, each piece must also
have an equal share of the perimeter of the square.**
If six people share the whole cake, then three people
will share half the cake. One way to divide the cake in
half is shown here.
Each half of the cake is 18 inches on its
two outer edges. So each person should receive 18/3, or
6 inches, of the cake’s outer edge. One way to make
this division is shown below:
To check the areas of the three pieces,
use the formula for the area of a triangle:
**Area = (1/2)
base ×
height**
The height of triangles 1 and 3 in the diagram
is half the width of the cake, or 4.5 inches. Since the
length of each base is 6 inches, the area of each of these
two triangles is:
**A = (1/2) ×
6 ×
4.5 = 13.5 in**^{2}
The area of shape 2 is half the area of
the entire cake, minus the area of triangles 1 and 3:
**Area of half
= (1/2) ×
9 ×
9 = 81/2**
= 40.5, or 40.5 in^{2}
**AreaTriangle1
= AreaTriangle3**
= (1/2) ×
6 ×
4.5 = 13.5, or 13.5 in^{2}
**AreaRegion2 is
Area of half of the cake**
– AreaTriangle1 – AreaTriangle3
**AreaRegion2 =
(1/2) ×
9 ×
9 –13.5 – 13.5 = 13.5, or 13.5 in**^{2}
Therefore, these three pieces represent
equal shares. The other half of the cake can be divided
similarly.
There are other ways to cut the cake into
six equal shares. For example, you could start by dividing
the cake into two rectangles of the same size and shape. |