Question

Find the perimeter and area of each rectangle. \(\ell=25 \mathrm{ft}, w=5 \mathrm{ft}\)

Short answer

The perimeter is 60 ft and the area is 125 sq ft.

Step-by-step solution

Understand the Problem

We need to find both the perimeter and the area of a rectangle given its length \(\ell = 25\, \mathrm{ft}\) and width \(w = 5\, \mathrm{ft}\).

Recall Perimeter Formula

The formula for the perimeter of a rectangle is given by \( P = 2\ell + 2w \). This formula is applicable to any rectangle.

Calculate the Perimeter

Substitute the given values into the formula: \( P = 2(25) + 2(5) \). Calculate inside the parentheses first, then multiply and add the results to get \( P = 50 + 10 = 60\, \mathrm{ft}\).

Recall Area Formula

The formula for the area of a rectangle is given by \( A = \ell \times w \). This formula calculates the surface inside the rectangle.

Calculate the Area

Substitute the given values into the area formula: \( A = 25 \times 5 \). Perform the multiplication to get \( A = 125\, \mathrm{ft}^2 \).

Key concepts

Rectangle Geometry

Rectangles are fundamental shapes in geometry characterized by having opposite sides that are equal in length and all four angles being right angles.
These properties make them a special type of parallelogram. When you look at a rectangle, one of the first things to notice is the length and the width, which are the measures of its longer and shorter sides, respectively.

• The **length** (\(\ell\)) is usually the longer side.
• The **width** (\(w\)) is typically the shorter side.

The shape's symmetry provides a straightforward way to compute both its perimeter and area, enabling us to easily handle calculations involving rectangles.

Mathematics Formulas

When it comes to calculating properties of rectangles, there are two primary formulas you'll want to remember:

• The **perimeter formula**: \( P = 2\ell + 2w \) - This formula allows us to find the distance around the rectangle, combining twice the length and width.
• The **area formula**: \( A = \ell \times w \) - This formula is used to calculate the space within the rectangle, by multiplying its length by its width.

These formulas are tools that help in various applications, from simple school problems to real-world tasks like flooring a room or framing a picture.

Measurement Calculations

Measurement calculations involving rectangles are direct applications of basic arithmetic.
For the perimeter, think about walking around the edge of the shape. You add up each side twice— once for the passage on one side and once for the passage on the opposite side, resulting in \( 2\ell + 2w \).
For the area, consider how much space is covered by the rectangle if it were laid flat. By multiplying the length by the width, \( \ell \times w \), you obtain the number of square units that fit inside.
Both these calculations require substituting the known values into each respective formula, as shown in the example problem with \( \ell = 25 \text{ ft}\) and \( w = 5 \text{ ft} \).
The solutions— \( P = 60 \text{ ft} \) for perimeter and \( A = 125 \text{ ft}^2 \) for area, illustrate a step-by-step process that helps solidify your understanding of practical geometry tasks.