Problem 70 - Elementary Algebra

Question

Each side of a square deck measures 8 feet. Determine the area and perimeter of the deck.

Step-by-Step Solution

Answer

The perimeter is 32 feet, and the area is 64 square feet.

Step 1: Understand the Problem

We are given a square deck with each side measuring 8 feet. We need to find both the area and the perimeter of this square.

Step 2: Recall the Formula for Perimeter

The perimeter of a square can be found using the formula: \( P = 4s \)where \( s \) is the length of one side of the square. In this case, \( s = 8 \) feet.

Step 3: Calculate the Perimeter

Substitute the side length into the perimeter formula:\[ P = 4 imes 8 = 32 \text{ feet} \]

Step 4: Recall the Formula for Area

The area of a square can be calculated using the formula: \( A = s^2 \)where \( s \) is the length of one side of the square. Again, \( s = 8 \) feet.

Step 5: Calculate the Area

Substitute the side length into the area formula:\[ A = 8^2 = 64 \text{ square feet} \]

Step 6: Review

We found the perimeter to be 32 feet and the area to be 64 square feet.

Key Concepts

Square Geometry

A square is a simple geometric shape with some unique properties that make its calculations straightforward. In geometry, a square is defined as a four-sided figure with all sides equal in length. Moreover, all the interior angles in a square are right angles, each measuring exactly 90 degrees.

Squares are special because they are regular quadrilaterals—meaning that not only are their sides equal, but the diagonals intersect at right angles and are of equal length too. While working with squares, once you know the length of one side, you have all the information you need for further calculations like area and perimeter.

Understanding these fundamental characteristics helps in visualizing the problem more vividly and prepares you for easy problem-solving when you encounter questions involving square geometry. Whether it’s calculating the perimeter or the area, knowing you’re dealing with a square allows you to directly apply specific formulas.

Perimeter Formula

The perimeter of a square is the total distance around the shape. Since all four sides of a square are equal, the formula for the perimeter is very simple. It's written as:

• \( P = 4s \)

where \( P \) represents the perimeter, and \( s \) is the length of one side of the square. This formula reflects that you are adding up four sides of equal length. In our given problem with sides each measuring 8 feet, the calculations become:

• \( P = 4 imes 8 = 32 \text{ feet} \)

By simply substituting the value of the side into the equation, you can quickly determine the perimeter of the square. This formula is especially handy because it applies universally to any square, regardless of the unit of measurement used. Knowing how to use the perimeter formula allows you to measure the total length of the boundary of the square easily.

Area Formula

The area of a square is a measure of the space contained within its boundaries. To calculate the area of a square, you use the formula:

• \( A = s^2 \)

Here, \( A \) represents the area, and \( s \) is the length of one side of the square. This formula is simply the side length multiplied by itself, as the area of a square is found by squaring the side length. In our example problem, each side is 8 feet long, leading to the area calculation:

• \( A = 8^2 = 64 \text{ square feet} \)

Calculating area using this formula is straightforward, once you know the side length, providing the total number of square units that fit within the square. Comprehending this formula enables you to visualize and calculate space efficiently, crucial for practical applications such as flooring, painting, or any area-based needs.

Problem Solving Steps

Finding the area and perimeter of a square requires specific steps. Approaching these problems methodically enhances accuracy and understanding. The outlined problem-solving steps include:

• Understand the Problem: Clearly identify the square's side length. In our case, each side is 8 feet.
• Recall Formulas: Memorize or access the perimeter formula \( P = 4s \) and the area formula \( A = s^2 \).
• Perform Calculations: Substitute the side length into each formula:
  • Perimeter: \( 4 imes 8 = 32 \text{ feet} \)
  • Area: \( 8^2 = 64 \text{ square feet} \)
• Review Results: Confirm that both the perimeter and area are correctly calculated values.

Following these steps guarantees a systematic approach to solving geometry problems involving squares. Practicing these steps repeatedly enhances your problem-solving skills over time, making it easier to tackle more complex mathematical concepts.