Recall that one of the ways to find the area of a rhombus is with the following formula:

Now, since all four sides in the rhombus are the same, we know from the given side value what the length of the base will be. In order to find the length of the height, we will need to use sine.

, where  is the given angle.

Now, plug this into the equation for the area to get the following equation:

Plug in the given side length and angle values to find the area.

Make sure to round to  places after the decimal.

To find the value of diagonal , we must first recognize some important properties of rhombuses. Since the perimeter is of  is , and by definition a rhombus has four sides of equal length, each side length of the rhombus is equal to . The diagonals of rhombuses also form four right triangles, with hypotenuses equal to the side length of the rhombus and legs equal to one-half the lengths of the diagonals. We can therefore use the Pythagorean Theorem to solve for one-half of the unknown diagonal:

, where  is the rhombus side length,  is one-half of the known diagonal, and  is one-half of the unknown diagonal. We can therefore solve for :

 is therefore equal to . Since  represents one-half of the unknown diagonal, we need to multiply by  to find the full length of diagonal .

The length of diagonal  is therefore 

This problem relies on the knowledge of the equation for the area of a rhombus, , where  is the area, and  and  are the lengths of the individual diagonals. We can substitute the values that we know into the equation to obtain:

Therefore, our final answer is that the diagonal 

The area of a rhombus is given below.

Substitute the given area and a diagonal.  Solve for the other diagonal.

The area of a rhombus is given below. Plug in the area and the given diagonal. Solve for the other diagonal.

Let the shorter diagonal be , and the longer diagonal be .  The longer dimension is twice as long as the other diagonal.  Write an expression for this.

Write the area of the rhombus.

Since we are solving for the shorter diagonal, it's best to setup the equation in terms , so that we can solve for the shorter diagonal.  Plug in the area and expression to solve for .

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the problem, we are given that the sides are  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the problem, we are given that the sides are  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the problem, we are given that the sides are  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  and use the Pythagorean Theorem to solve for . From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Using the quadratic formula,

With this equation, we get two solutions:

Only the positive solution is valid for this problem.