Problem 47 - Algebra 2 and Trigonometry

Question

The area of a square is 14 square centimeters. What is the length of a side of the square?

Step-by-Step Solution

Answer

The length of a side of the square is approximately 3.74 centimeters.

Step 1: Understand the Area Formula

The formula for the area of a square is given by \( A = s^2 \), where \( s \) is the length of a side of the square.

Step 2: Set Up the Equation

Set up the equation using the provided area. Since the area \( A \) is 14 square centimeters, the equation is \( 14 = s^2 \).

Step 3: Solve for the Side Length

To find the side length \( s \), take the square root of both sides of the equation. Thus, \( s = \sqrt{14} \).

Step 4: Approximate the Square Root

Using a calculator, evaluate \( \sqrt{14} \). The approximate value is \( 3.74 \) (rounded to two decimal places).

Key Concepts

Area of a Square

To understand how to find the side length when given the area of a square, it's essential to grasp the concept of area itself. The area measures the space inside a two-dimensional shape.
The specific formula for the area of a square is simple yet powerful:

• \( A = s^2 \)

Here, \( A \) represents the area, and \( s \) is the length of one side of the square. Each side of a square is equal in length, which simplifies calculations. This formula derives from multiplying the length of the side by itself, signifying the squared nature of a square. If you know any one of these two values, you can solve for the other. In the case of the exercise, the area is known, and we solve for the side length.

Solving Quadratic Equations

When working with the square's area formula, solving for a side length becomes a problem of solving a quadratic equation. A quadratic equation follows the form \( ax^2 + bx + c = 0 \). In geometric contexts like this exercise, it typically simplifies to \( s^2 = A \), where we'll solve for \( s \).
Here’s a quick process for solving such an equation:

• Rearrange the equation to set the quadratic term on one side.
• Isolate \( s^2 \) (our squared variable).
• Take the square root of both sides to solve for \( s \).

Square roots are the mathematical operation that "undoes" a square; thus, it's the perfect tool here. For instance, to find \( s \) in our exercise, compute \( s = \sqrt{14} \). Understanding the steps not only aids in geometry but also introduces a core algebraic method used in various problems.

Geometry Formulas

Geometry involves numerous formulas that describe and measure different shapes and their properties. Here are a few foundational formulas relevant to squares and similar problems:

• Perimeter of a square: \( P = 4s \)
• Diagonal of a square: \( d = s\sqrt{2} \)

Each of these formulas employs the side length \( s \), showcasing the interlinked nature of geometry. Knowing one measurement, such as the area, allows for the calculation of other properties using these relationships. By understanding and applying these formulas, you can quickly navigate and solve a wide array of geometric problems efficiently. Remember, geometry often provides not just numeric answers, but insights into spatial reasoning and logic. Each formula is a tool in the toolkit for approaching and solving the puzzles of space and dimension.