Area and perimeter worksheet featuring grid-based problems for drawing and calculating rectangle areas.

Problem Analysis:

The worksheet involves tasks related to the area and perimeter of rectangles. The key concepts are:
1. Area of a Rectangle: The area is calculated as \( \text{Area} = \text{length} \times \text{width} \).
2. Drawing Rectangles: Students need to draw rectangles on a grid based on given areas.
3. Finding Areas: Students need to calculate the area of given rectangles.

Let's solve each part step by step.

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Section 1: Draw a rectangle with an area of 42 square units (1a)

#### Solution:
To draw a rectangle with an area of 42 square units, we need to find pairs of length and width whose product is 42. Possible pairs are:
- \( 6 \times 7 \)
- \( 7 \times 6 \)
- \( 1 \times 42 \)
- \( 42 \times 1 \)
- \( 2 \times 21 \)
- \( 21 \times 2 \)
- \( 3 \times 14 \)
- \( 14 \times 3 \)

Choose one pair, for example, \( 6 \times 7 \). Draw a rectangle with 6 units in length and 7 units in width on the grid.

#### Answer:
Draw a rectangle with dimensions \( 6 \times 7 \).

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Section 1b: Find the area of this rectangle

#### Solution:
The rectangle in Section 1b is already drawn. Count the number of squares it occupies:
- The rectangle has 4 rows and 5 columns.
- Therefore, the area is \( 4 \times 5 = 20 \) square units.

#### Answer:
The area is \( 20 \) square units.

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Section 2a: Find the area of this rectangle

#### Solution:
The rectangle in Section 2a is already drawn. Count the number of squares it occupies:
- The rectangle has 1 row and 6 columns.
- Therefore, the area is \( 1 \times 6 = 6 \) square units.

#### Answer:
The area is \( 6 \) square units.

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Section 2b: Draw a rectangle with an area of 40 square units

#### Solution:
To draw a rectangle with an area of 40 square units, we need to find pairs of length and width whose product is 40. Possible pairs are:
- \( 5 \times 8 \)
- \( 8 \times 5 \)
- \( 1 \times 40 \)
- \( 40 \times 1 \)
- \( 2 \times 20 \)
- \( 20 \times 2 \)
- \( 4 \times 10 \)
- \( 10 \times 4 \)

Choose one pair, for example, \( 5 \times 8 \). Draw a rectangle with 5 units in length and 8 units in width on the grid.

#### Answer:
Draw a rectangle with dimensions \( 5 \times 8 \).

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Section 3a: Draw a rectangle with an area of 45 square units

#### Solution:
To draw a rectangle with an area of 45 square units, we need to find pairs of length and width whose product is 45. Possible pairs are:
- \( 5 \times 9 \)
- \( 9 \times 5 \)
- \( 1 \times 45 \)
- \( 45 \times 1 \)
- \( 3 \times 15 \)
- \( 15 \times 3 \)

Choose one pair, for example, \( 5 \times 9 \). Draw a rectangle with 5 units in length and 9 units in width on the grid.

#### Answer:
Draw a rectangle with dimensions \( 5 \times 9 \).

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Section 3b: Draw a rectangle with an area of 28 square units

#### Solution:
To draw a rectangle with an area of 28 square units, we need to find pairs of length and width whose product is 28. Possible pairs are:
- \( 4 \times 7 \)
- \( 7 \times 4 \)
- \( 1 \times 28 \)
- \( 28 \times 1 \)
- \( 2 \times 14 \)
- \( 14 \times 2 \)

Choose one pair, for example, \( 4 \times 7 \). Draw a rectangle with 4 units in length and 7 units in width on the grid.

#### Answer:
Draw a rectangle with dimensions \( 4 \times 7 \).

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Final Answers:

1a. Draw a rectangle with dimensions \( 6 \times 7 \).
1b. The area is \( 20 \) square units.
2a. The area is \( 6 \) square units.
2b. Draw a rectangle with dimensions \( 5 \times 8 \).
3a. Draw a rectangle with dimensions \( 5 \times 9 \).
3b. Draw a rectangle with dimensions \( 4 \times 7 \).

\[
\boxed{
\begin{array}{l}
\text{1a. } 6 \times 7 \\
\text{1b. } 20 \\
\text{2a. } 6 \\
\text{2b. } 5 \times 8 \\
\text{3a. } 5 \times 9 \\
\text{3b. } 4 \times 7
\end{array}
}
\]