We begin this problem by finding the difference of two areas: the larger rectangle bounded by the outer edge of the border and the smaller rectangle that is the court itself.

The larger rectangle is  square feet , and the court is  square feet .

The difference, , repesents the area of the border.

Now we divide this by , which is just a bit over . But since we can't leave  square feet unpainted, we have to round up to  cans of paint.

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces): 

Left, right (two surfaces): 

The total area: 

Since a square has four sides of equal length, the solid looks like this:

The areas of each of the individual surfaces, each of which is a rectangle, are the product of their dimensions:

Front, back, top, bottom (four surfaces): 

Left, right (two surfaces): 

The total surface area is therefore 

A rhombus, by definition, has four sides of equal length. Therefore, , and, by the Multiplication Property, . Also, since  and  are the midpoints of their respective sides,  and . Combining these statements, and letting :

Also, both  and  are altitudes of the rhombus; they are congruent, and we will call their common length  (height).

The figure, with the lengths, is below.

The area of the entire Rhombus  is the product of its height  and the length of a base , so

.

 Rectangle  has as its length and width  and , so its area is their product , Since

,

From the Division Property, it follows that

,

and

.

This makes 450 the area of Rectangle . 

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. So we can write:

Base length is so the corresponding altitude is  .

The area of a parallelogram is given by:

Where:

is the length of any base
is the corresponding altitude

So we can write:

Let  be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to , 40% of  is equal to .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

Let  be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for :

To convert yards to inches, multiply by 36:

Let  be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is 

of , or .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

A quadrilateral is any two-dimensional shape with   sides. The only shape listed that does not have  sides is a triangle.