Question

A square has an area of 64 square centimeters. What is the length of its sides? A. 4 centimeters B. 8 centimeters C. 16 centimeters D. 32 centimeters

Short answer

The length of the sides is 8 centimeters, option B.

Step-by-step solution

Identify the formula for the area of a square

The area of a square can be calculated using the formula: \[ \text{Area} = \text{side}^2 \] where 'side' represents the length of one side of the square.

Set up the equation

Using the given area, set up the equation based on the formula: \[ 64 = \text{side}^2 \]

Solve for the length of the side

To find the length of the side, take the square root of both sides of the equation: \[ \text{side} = \sqrt{64} \]\[ \text{side} = 8 \]

Verify the answer with the given options

Compare the calculated side length to the given options. The matching option is B, 8 centimeters.

Key concepts

Understanding the Geometry of a Square

In geometry, a square is a special type of polygon that has four equal sides and four right angles. Essentially, every side of a square is of the same length. Due to its symmetrical properties, understanding a square is foundational to many areas of mathematics. When calculating parameters like the perimeter and area, these properties simplify the equations. For example, ensuring each side is equal makes it easier to identify that the area of a square is obtained by squaring the length of one of its sides.

Knowing these properties also allows us to easily solve problems dealing with the area and sides of squares by applying straightforward formulas. This makes the square both a simple and powerful shape in geometry.

The Role of Square Roots in Solving Area Problems

A square root is a mathematical concept that essentially reverses the process of squaring a number. It finds the original number that was multiplied by itself to reach the given value.

For instance, finding the side length of a square when the area is given requires taking the square root of the area. In our exercise, the area of the square is 64 square centimeters, meaning we must determine what number, when squared, equals 64. This is found by computing the square root of 64 which is 8. \( \text{side} = \sqrt{64} = 8 \).

Square roots are very useful in various branches of mathematics and science, helping solve equations dealing with areas and volumes where one dimension is unknown.

Using Mathematical Reasoning to Verify Solutions

Mathematical reasoning involves applying logical steps to solve problems and verify answers. This is crucial to understanding and confirming our results. In this exercise, we verified our solution by following logical steps:

• We began by recognizing the area formula for a square: \( \text{Area} = \text{side}^2 \).
• Next, we substituted the given area into the formula: 64 = \text{side}^2 \.
• We then solved for the side length by taking the square root of both sides, finding that \text{side} = \sqrt{64} = 8 \
.
• Finally, we compared our calculated side length to the options given, confirming our solution matched option B (8 centimeters).

This approach of breaking problems into systematic steps and verifying each step ensures accuracy and reinforces understanding of the foundational concepts.