Step 1: Understand the Key Formulas
There are two main formulas we'll use:
1. Perimeter formula for a parallelogram:
\[ P = 2(a + b) \]
2. Area formula for a parallelogram:
\[ A = a \times b \times \sin(\theta) \]
where \( \theta \) is the angle between the sides \( a \) and \( b \).
Step 2: Use Actual Numbers
Let's assume:
- The perimeter (\(P\)) is 36 units.
- One side length (\(a\)) is 10 units.
- The angle (\(\theta\)) between the sides is 30 degrees.
Step 3: Calculate the Length of the Other Side
First, we need to find the length of the other side (\(b\)). Using the perimeter formula:
\[ P = 2(a + b) \]
Substitute the given values:
\[ 36 = 2(10 + b) \]
Divide both sides by 2:
\[ 18 = 10 + b \]
Solve for \(b\):
\[ b = 18 - 10 \]
\[ b = 8 \, \text{units} \]
Step 4: Convert the Angle to Radians
Since trigonometric functions in the area formula use radians, convert the angle from degrees to radians:
\[ \theta = 30^\circ = \frac{30 \times \pi}{180} = \frac{\pi}{6} \]
Step 5: Calculate the Area
Now, use the area formula with the calculated side lengths and angle:
\[ A = a \times b \times \sin(\theta) \]
Substitute the known values:
\[ A = 10 \times 8 \times \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Therefore:
\[ A = 10 \times 8 \times \frac{1}{2} \]
Perform the multiplication:
\[ A = 10 \times 8 \times 0.5 \]
\[ A = 40 \, \text{square units} \]
Summary of Steps
1. **Understand the formulas**: \( P = 2(a + b) \) and \( A = a \times b \times \sin(\theta) \)
2. **Use real whole numbers**: \( P = 36 \) units, \( a = 10 \) units, \( \theta = 30^\circ \)
3. **Calculate the other side**: \( b = 8 \) units
4. **Convert the angle to radians**: \( \theta = \frac{\pi}{6} \)
5. **Calculate the area**: \( A = 40 \) square units
By following these steps, you can easily determine the area of a parallelogram given the perimeter, one side length, and the angle between them.
