Calculating the Rhombus Perimeter from Area and Angle

Understanding the Given Information

We are asked to find the perimeter of a rhombus. We are provided with two key pieces of information about this rhombus:

• The area of the rhombus is $10\text{ cm}^2$.
• One of its interior angles is $150^{\circ}$.

To find the perimeter, we first need to determine the length of one side of the rhombus, as all sides of a rhombus are equal.

Rhombus Area Formula Explained

The area of a rhombus can be calculated using its side length and one of its interior angles. The formula is:

$Area = s^2 \sin(\theta)$

In this formula:

• $s$ represents the length of a side of the rhombus.
• $\theta$ represents one of the interior angles of the rhombus.

Step 1: Applying the Area Formula

Let's substitute the given values into the area formula:

• Area = $10\text{ cm}^2$
• \theta = $150^{\circ}$

So, the equation becomes:

$10 = s^2 \sin(150^{\circ})$

Step 2: Determining the Sine of the Angle

We need to find the value of $\sin(150^{\circ})$. We can use the property that $\sin(180^{\circ} - x) = \sin(x)$.

Therefore:

$\sin(150^{\circ}) = \sin(180^{\circ} - 30^{\circ}) = \sin(30^{\circ})$

The value of $\sin(30^{\circ})$ is a standard trigonometric value:

$\sin(30^{\circ}) = \frac{1}{2}$

So, $\sin(150^{\circ}) = \frac{1}{2}$.

Step 3: Calculating the Side Length (s)

Now, substitute the value of $\sin(150^{\circ})$ back into our area equation:

$10 = s^2 \times \frac{1}{2}$

To find $s^2$, we multiply both sides of the equation by 2:

$10 \times 2 = s^2$

$20 = s^2$

To find the side length 's', we take the square root of both sides:

$s = \sqrt{20}$

We can simplify $\sqrt{20}$ by finding perfect square factors:

$s = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$

The length of one side of the rhombus is $2\sqrt{5}$ cm.

Step 4: Calculating the Perimeter

The perimeter of a rhombus is calculated by multiplying the side length by 4, since all four sides are equal.

$Perimeter = 4 \times s$

Using the side length we found:

Perimeter = $4 \times (2\sqrt{5})$

Perimeter = $8\sqrt{5}$ cm

Final Answer

The perimeter of the rhombus is $8\sqrt{5}$ cm.