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To solve this problem, let's break it down based on the diagram and the details provided.

We are asked to find the area of trapezoid \( ABCD \), and it is given that quadrilateral \( PBCD \) is a rectangle with an area of 24 cm².

### Step-by-step process:

1. **Identifying Key Elements:**
   - Trapezoid \( ABCD \) has bases \( AB \) and \( CD \).
   - \( AB = 5 \, \text{cm} \) and \( CD = 6 \, \text{cm} \), as shown in the diagram.
   - \( PBCD \) is a rectangle, with one side as \( PB \) (which is equal to \( AB \), since it's opposite in a rectangle).

2. **Dimensions of Rectangle \( PBCD \):**
   - The area of \( PBCD \) is given as \( 24 \, \text{cm}^2 \). The length of \( CD = 6 \, \text{cm} \).
   - Using the formula for the area of a rectangle \( \text{Area} = \text{base} \times \text{height} \), we can find the height \( h \) of the rectangle:
     \[
     \text{Area} = 6 \times h = 24
     \]
     Solving for \( h \):
     \[
     h = \frac{24}{6} = 4 \, \text{cm}
     \]
     So, the height of both the rectangle and trapezoid is \( 4 \, \text{cm} \).

3. **Calculating the Area of Trapezoid \( ABCD \):**
   - The area of a trapezoid is given by:
     \[
     \text{Area of trapezoid} = \frac{1}{2} \times ( \text{Base}_1 + \text{Base}_2 ) \times \text{Height}
     \]
     Here, \( \text{Base}_1 = 5 \, \text{cm} \), \( \text{Base}_2 = 6 \, \text{cm} \), and \( \text{Height} = 4 \, \text{cm} \).

   - Substituting these values into the formula:
     \[
     \text{Area} = \frac{1}{2} \times (5 + 6) \times 4
     \]
     \[
     \text{Area} = \frac{1}{2} \times 11 \times 4 = 22 \, \text{cm}^2
     \]

Thus, the area of trapezoid \( ABCD \) is \( \boxed{22 \, \text{cm}^2} \).

Would you like more details or have any questions?

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### Related Questions:
1. What is the formula for the area of a trapezoid, and how does it differ from the area of a rectangle?
2. How would the area of trapezoid \( ABCD \) change if the height were doubled?
3. If the base \( AB \) were 7 cm instead of 5 cm, how would that affect the area?
4. How do you determine the height of a shape given its area and base dimensions?
5. Can you apply the same method to find the area of a parallelogram?

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**Tip:** Always ensure that you correctly identify the height in problems involving trapezoids and rectangles, as it is crucial for calculating the area.

Calculate the area of trapezoid ABCD, if PBCD is a rectangle with an area of 24 cm².

Learn how to calculate the area of trapezoid ABCD when given that quadrilateral PBCD is a rectangle with an area of 24 cm². This problem involves key geometric concepts such as the area of a rectangle and the area of a trapezoid. Suitable for students in Grades 6-8.

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