Introduction
Calculating the area of a rhombus is essential in geometry and various real-world applications. When you know the length of one side \( a \) and one of the inner angles, you can easily find the area using specific formulas and strategies. In this guide, we'll explore the steps to find the area of a rhombus in such a scenario.
Understanding the Rhombus
A rhombus is a quadrilateral with all four sides of equal length. Opposite angles in a rhombus are equal, but unlike rectangles or squares, its angles are not necessarily right angles.
The Formula for the Area of a Rhombus
The area \( A \) of a rhombus can be found using the formula:
\[ A = a^2 \sin(\theta) \]
Where:
- \( a \) is the length of one side of the rhombus.
- \( \theta \) is one of the inner angles of the rhombus.
Explaining the Formula
The area formula for a rhombus involves multiplying the square of the length of one side \( a \) by the sine of one of the inner angles \( \theta \). This formula exploits the trigonometric relationship between the side length and the angles of the rhombus.
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a rhombus with a side length \( a = 8 \) units and an inner angle \( \theta = 60^\circ \). We want to find the area of the rhombus.
Step 1: Identify the Given Values
Given:
- Side length \( a = 8 \) units
- Inner angle \( \theta = 60^\circ \)
Step 2: Use the Formula to Find the Area
Using the formula \( A = a^2 \sin(\theta) \), substitute the given values:
\[ A = 8^2 \times \sin(60^\circ) \]
Step 3: Perform the Calculation
Now, calculate the area:
\[ A = 64 \times \sin(60^\circ) \]
\[ A = 64 \times 0.866 \]
\[ A \approx 55.104 \]
Final Value
For a rhombus with a side length \( a = 8 \) units and an inner angle \( \theta = 60^\circ \), the area is approximately \( 55.104 \) square units.
