Problem 60 - Prealgebra

Question

Find the area and perimeter of each square if the length of each side is as given below. \(s=14\) yards

Step-by-Step Solution

Answer

The area is 196 square yards, and the perimeter is 56 yards.

Step 1: Identify the Formula for Area

The area \(A\) of a square with side length \(s\) is found using the formula \(A = s^2\).

Step 2: Calculate the Area

Substitute \(s = 14\) yards into the formula: \[ A = 14^2 = 196 \text{ square yards} \]Thus, the area of the square is 196 square yards.

Step 3: Identify the Formula for Perimeter

The perimeter \(P\) of a square with side length \(s\) is found using the formula \(P = 4s\).

Step 4: Calculate the Perimeter

Substitute \(s = 14\) yards into the formula: \[ P = 4 \times 14 = 56 \text{ yards} \]Thus, the perimeter of the square is 56 yards.

Key Concepts

Understanding the Area of a Square

The area of a square is an important concept in geometry. It measures the size of a surface, or how much space a flat shape like a square takes up on a plane. To find the area, we rely on a simple mathematical formula. For a square, this formula is based on the length of its sides.

The formula to compute the area of a square is given by the equation:

• \( A = s^2 \)

where \( A \) represents the area, and \( s \) is the length of one of the square's sides.

Understanding this, when you know the side length of a square, you simply multiply it by itself to find the area. For instance, if each side of the square is 14 yards, then the area is calculated as \( 14 \times 14 \). This comes out to \( 196 \) square yards. The **unit** for area is always a "squared" unit, such as square yards, square meters, etc., which shows the two-dimensional nature of the measurement.

Exploring the Perimeter of a Square

The perimeter of a square is a measure of the total length around it. Imagine walking around the outer edge of a square, and measuring the distance you've traveled by the time you return to your starting point. That's essentially the perimeter.

To compute it, we use a straightforward mathematical formula:

• \( P = 4s \)

This equation represents the idea that a square has four sides of equal length. Hence, multiplying the side length \( s \) by 4 gives us the perimeter.

Consider a square where each side is 14 yards long. By applying the formula, \( P = 4 \times 14 \), we determine the perimeter to be \( 56 \) yards. It’s important to note that the unit for the perimeter will be the same as the unit for its side length, in this case, yards.

Decoding Mathematical Formulas

Mathematical formulas are essential tools in geometry as they offer rules and equations to solve various problems. Understanding how to use them correctly is key to success.

**Components of a Mathematical Formula**

• Variables: Symbols representing quantities that can change.
• Constants: Fixed values within an equation.
• Operations: Mathematical processes like addition or multiplication.

For instance, in the formula for the area of a square \( A = s^2 \), \( A \) is the variable, \( s \) is both a constant and a variable (depending on its use in context), and \(^2\) represents multiplication.

Whenever applying a formula, follow these steps:

• Identify given values and the formula needed.
• Substitute numbers into the formula in place of variables.
• Perform the necessary calculations.
• Check the unit of your answer.

This structured approach ensures precision and aids in avoiding mistakes, leading to confident problem-solving in geometry.