Question

Find the perimeter and area of each rectangle described. \(\ell=18 \mathrm{~cm}, w=18 \mathrm{~cm}\)

Short answer

The perimeter is 72 cm and the area is 324 cm².

Step-by-step solution

Identify the rectangle formulas

To find the perimeter and area of a rectangle, we use the formulas: - Perimeter ( P ) of a rectangle: P = 2( ext{length} + ext{width} ) - Area ( A ) of a rectangle: A = ext{length} imes ext{width} .

Substitute the given values into the perimeter formula

We know the length ( ext{l}) is 18 cm and the width ( ext{w}) is also 18 cm. Plugging these into the perimeter formula gives:\[ P = 2(18 + 18) = 2 imes 36 = 72 ext{ cm} \]

Substitute the given values into the area formula

Using the same values for length and width, substitute them into the area formula:\[ A = 18 imes 18 = 324 ext{ cm}^2 \]

Key concepts

Perimeter Formula

The perimeter of a rectangle is a measure that represents the total distance around the outer edge of the rectangle. To calculate it, we use the following formula: \[P = 2(\text{length} + \text{width})\]This formula tells us to double the sum of the length and the width.

This is because a rectangle has two pairs of equal and opposite sides. - One pair is made of the two lengths.- The other pair is composed of the two widths.Therefore, by adding the length and width together and multiplying by two, you account for all sides.
In a rectangle where both the length and width are 18 cm, like a square, the calculation would be:\[P = 2(18 + 18) = 2 \times 36 = 72 \text{ cm}\]So, the perimeter is 72 cm. This method ensures you can find the perimeter for any rectangle, no matter its size.

Area Formula

The area of a rectangle represents the amount of two-dimensional space taken up by the rectangle. To find the area, you use the formula:\[A = \text{length}\times \text{width}\]This formula essentially tells us to multiply the length by the width.

Each side is considered in a pair with the other side. - The length represents one dimension of the rectangle. - The width represents the other.Together, they allow you to calculate the full extent of space inside the rectangle.
For our rectangle, where both the length and width are 18 cm, the formula becomes:\[A = 18 \times 18 = 324 \text{ cm}^2\]Thus, the area is 324 square centimeters. This formula can be used universally to determine the area of any rectangle you encounter.

Rectangle Properties

Rectangles are one of the most common shapes you encounter in geometry. They have specific properties that distinguish them from other shapes: - **Four Right Angles:** Each angle in a rectangle is 90 degrees. This is what makes its opposite sides parallel and equal. - **Opposite Sides Are Equal:** In a rectangle, opposite sides are equal in length. This means that the length equals the length of the opposite side, and the width equals the width of the opposite side.
These properties are crucial because they ensure the rectangle remains a consistent shape. - **Diagonal Properties:** The diagonals of a rectangle are equal in length, another feature that comes from the properties of its shape. In our problem, since the length and width are equal (both are 18 cm), - The rectangle is actually a square. - Every side is the same length. - A rectangle that is also a square has even more symmetrical properties. Understanding these traits helps in recognizing and solving problems involving rectangles in geometry.