One possible shape is a twenty-sided die, known as an icosahdron. The formula for its volume is . After substituting 500 for V, a is found to be approximately

6.1196

. This is then substituted into the formula for surface area, . This yields a result of

324.3222 cm²

.



Another alternative is a cube. Its volume is defined as

V=s³

. When solved with V=500, s comes to

7.937 cm

. When it is plugged into the formula

SA=6s²

, the surface area comes to

378 cm³

.






If you were hoping for a more unique and distinguishable container, a pyramid or a cone would be a good choice. For both of these options, the simple formula for volume is . In solving for a square pyramid, it makes the most sense to leave b as the area of the entire base. Also, height must be rewritten in terms of b, which is . Since the surface area requires the slant height to determine the area of the triangles, that also needs to be found. The final equation is . The graph of this looks like this:

The optimal point has a base area of

104.0042 cm²

, therefore a base side length of

10.1982 cm

, a height of

14.4225 cm

, and a surface area of

416.0168 cm²

.

With a cone, it makes more sense to use

pi*r²

for b. This gives a height of and a surface area equation of with the graph:

At its optimal point, the cone has a base radius of

5.5267 cm

, a height of

15.6318 cm

, and a surface area of

383.8317 cm²

.