Rhombus Perimeter Calculation Using Area and Angle

This solution demonstrates how to calculate the perimeter of a rhombus when provided with its area and the measure of one of its interior angles.

Rhombus Properties Overview

A rhombus is a quadrilateral characterized by four equal side lengths. Key properties relevant to this problem include:

• All four sides are equal. Let the side length be denoted by '$a$'.
• The perimeter ($P$) is calculated as $P = 4a$.
• The area can be determined using the formula involving a side and an angle.

Area Calculation Formula

The area of a rhombus can be expressed using the formula:

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

Here, '$a$' represents the length of a side of the rhombus, and '$\theta$' is one of the interior angles.

Applying Given Information

The problem provides the following details:

• Area $= 10$ cm$^2$
• An interior angle $\theta = 150°$

The objective is to determine the rhombus's perimeter.

Step 1: Determining the Side Length ($a$)

We can find the side length '$a$' by rearranging the area formula.

First, let's find the value of $\sin(150°)$. Using trigonometric identities:

$\sin(150°) = \sin(180° - 30°) = \sin(30°) = \frac{1}{2}$

Now, substitute the known values into the area formula:

$10 \text{ cm}^2 = a^2 \times \sin(150°)$

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

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

$a^2 = 10 \times 2$

$a^2 = 20$

Next, calculate the side length '$a$' by taking the square root of $a^2$:

$a = \sqrt{20}$ cm

Simplify the radical:

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

Step 2: Calculating the Perimeter ($P$)

With the side length '$a$' calculated as $2\sqrt{5}$ cm, we can now find the perimeter using the formula $P = 4a$.

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

$P = 8\sqrt{5}$ cm

Final Result

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