Question

Find the perimeter and area of each rectangle. \(\ell=18 \mathrm{~cm}, w=12 \mathrm{~cm}\)

Short answer

Perimeter: 60 cm, Area: 216 cm².

Step-by-step solution

Understand the given dimensions

The rectangle is specified with a length \( \ell = 18 \mathrm{~cm} \) and a width \( w = 12 \mathrm{~cm} \). Identify these as the dimensions needed for calculating perimeter and area.

Calculate the perimeter

The perimeter \( P \) of a rectangle is given by the formula \( P = 2(\ell + w) \). Substitute the given dimensions into the formula: \[ P = 2(18 \mathrm{~cm} + 12 \mathrm{~cm}) = 2 \times 30 \mathrm{~cm} = 60 \mathrm{~cm} \].

Calculate the area

The area \( A \) of a rectangle is given by the formula \( A = \ell \times w \). Use the given dimensions:\[ A = 18 \mathrm{~cm} \times 12 \mathrm{~cm} = 216 \mathrm{~cm}^2 \].

Key concepts

Understanding the Perimeter

The perimeter of a rectangle is simply the total length you would walk if you traced the outer boundary of the shape. It's a measure of the boundary's length. When you are asked to find the perimeter of a rectangle, you are looking to sum up the lengths of all four sides of the rectangle. However, since a rectangle has two pairs of equal sides, you can make this calculation easier.

To find the perimeter, use the formula: \[ P = 2(\ell + w) \]
where \( \ell \) is the length and \( w \) is the width. This formula works because you are adding the length to the width and then doubling it to account for both pairs of sides.

• Substitute the given dimensions: If \( \ell = 18 \text{ cm} \) and \( w = 12 \text{ cm} \), then \[ P = 2(18 \text{ cm} + 12 \text{ cm}) = 60 \text{ cm} \].
• This means that if you walked around the entire rectangle, you would cover a distance of 60 centimeters.

Calculating the Area

The area of a rectangle is a measure of how much space is contained within its four sides. Think of it as the amount of surface you could cover, like how many tiles fit perfectly within the rectangle. Calculating area is essential in many real-life situations, such as determining the amount of paint needed to cover a wall.

The formula to find the area is: \[ A = \ell \times w \]
Simply multiply the length by the width to get the total area.

• Using our given dimensions where \( \ell = 18 \text{ cm} \) and \( w = 12 \text{ cm} \), the area calculation is \[ A = 18 \text{ cm} \times 12 \text{ cm} = 216 \text{ cm}^2 \].
• This means the space enclosed by the rectangle is 216 square centimeters, which could help in visualizing the coverage area.

Defining Length and Width

Length and width are key dimensions of a rectangle. These measurements define the size and shape of the rectangle.

• Length (\( \ell \)): This is the longer side of the rectangle. In our example, the length is 18 cm.
• Width (\( w \)): Also known as breadth, this is the shorter side of the rectangle, measuring 12 cm in our case.

Understanding which dimension is which can help prevent errors in calculations, especially important when both measurements are needed, such as in calculating area or perimeter.

Remember that the convention of labeling sides as length and width is merely to distinguish between dimensions in problems and solutions. It helps provide a better understanding and communication of the size and properties of a rectangle.