Rhombus Area and Perimeter Calculation

This solution explains how to find the perimeter of a rhombus when its area and one of its interior angles are given.

Understanding Rhombus Properties

A rhombus is a quadrilateral with all four sides equal in length. Let the side length of the rhombus be denoted by $a$. The perimeter of a rhombus is calculated as $4a$.

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

$ \text{Area} = a^2 \sin(\theta) $

where $a$ is the side length and $θ$ is one of the interior angles.

Applying the Area Formula

We are given:

• Area = $10 \text{ cm}^2$
• One interior angle ($\theta$) = $150^\circ$

Substitute these values into the area formula:

$ 10 = a^2 \sin(150^\circ) $

To solve for $a$, we first need the value of $\sin(150^\circ)$. We know 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 $\frac{1}{2}$.

So, the equation becomes:

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

Calculating Side Length

Now, we solve the equation for $a^2$:

$ a^2 = 10 \times 2 $

$ a^2 = 20 $

To find the side length $a$, we take the square root of $a^2$:

$ a = \sqrt{20} $

We can simplify $\sqrt{20}$:

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

So, the side length of the rhombus is $2\sqrt{5}$ cm.

Determining the Perimeter

The perimeter of a rhombus is 4 times its side length ($4a$).

$ \text{Perimeter} = 4 \times a $

Substitute the value of $a$ we found:

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

$ \text{Perimeter} = 8\sqrt{5} \text{ cm} $

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