Calculating the perimeter of a parallelogram ABCD can be done using trigonometry when the height \( h \) between sides \( AB \) and \( CD \) and the angle \( \angle ADC \) are given. This article will guide you through the process using the formula \( P = 2 \cdot (AB + AD) \), where \( AB = DC \) and \( AD = BC \). We will explain the formula and provide a step-by-step example to illustrate the calculations.
The Formula for the Perimeter of a Parallelogram
The perimeter \( P \) of a parallelogram is calculated by:
\[ P = 2 \cdot (AB + AD) = 2 \cdot AB + 2 \cdot AD \]
Where:
- \( P \) is the perimeter of the parallelogram.
- \( AB \) and \( DC \) are the lengths of one pair of opposite sides.
- \( AD \) and \( BC \) are the lengths of the other pair of opposite sides.
Explanation of the Formula
\( 2 \cdot (AB + AD) \): This part of the formula adds the lengths of adjacent sides \( AB \) and \( AD \), then multiplies by 2 to account for both pairs of opposite sides.
\( = 2 \cdot AB + 2 \cdot AD \): This is an alternative way to express the same calculation, showing the perimeter as the sum of twice the lengths of both pairs of opposite sides.
Using Trigonometry to Determine \( AD \)
To find the length of \( AD \), we use the given height \( h \) and the angle \( \angle ADC \). The relationship between these values is given by:
\[ AD = \dfrac{h}{\sin(\theta)} \]
where \( \theta \) is the angle \( \angle ADC \).
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a parallelogram ABCD with:
- Side \( AB = 10 \) units
- Height \( h = 6 \) units
- Angle \( \angle ADC = 30^\circ \)
We want to find the perimeter of the parallelogram.
Step 1: Identify the Given Values
Given:
- \( AB = 10 \) units
- Height \( h = 6 \) units
- \( \angle ADC = 30^\circ \)
Step 2: Calculate \( AD \) Using Trigonometry
Using the formula:
\[ AD = \dfrac{h}{\sin(\theta)} \]
Substitute the given values:
\[ AD = \dfrac{6}{\sin(30^\circ)} \]
Since \( \sin(30^\circ) = 0.5 \):
\[ AD = \dfrac{6}{0.5} = 12 \text{ units} \]
Step 3: Substitute the Values into the Perimeter Formula
\[ P = 2 \cdot (AB + AD) = 2 \cdot (10 + 12) \]
Step 4: Calculate the Perimeter
First, calculate the sum inside the parentheses:
\[ 10 + 12 = 22 \]
Then, multiply by 2:
\[ 2 \cdot 22 = 44 \]
Final Value
For a parallelogram with side \( AB = 10 \) units, height \( h = 6 \) units, and angle \( \angle ADC = 30^\circ \), the perimeter is 44 units.
Using trigonometry to find the length of \( AD \) and then calculating the perimeter ensures accurate results, making this method practical for various applications.
