Cube Edge Length Calculation If Lateral Surface Area Is 100 Square Inches

Jul 11, 2025 by ADMIN 74 views

If the Lateral Surface Area of a Cube Is 100 Square Inches, What Is the Length of an Edge?

In the world of geometry, cubes hold a special place due to their symmetrical and straightforward nature. Understanding the properties of a cube, such as its surface area and edge length, is fundamental in various mathematical and real-world applications. This article delves into a classic problem involving the lateral surface area of a cube and aims to determine the length of its edge. Let's explore the concepts, formulas, and step-by-step solution to unravel this geometric puzzle.

Before we dive into the problem, let's refresh our understanding of a cube and its key properties. A cube is a three-dimensional solid object bounded by six square faces, with three meeting at each vertex. All the edges of a cube have the same length, making it a highly symmetrical shape. The surface area of a cube is the total area of all its faces, while the lateral surface area refers to the sum of the areas of its four side faces, excluding the top and bottom faces.

Key Properties of a Cube

Faces: A cube has six faces, all of which are squares.
Edges: A cube has twelve edges, all of equal length.
Vertices: A cube has eight vertices, where three edges meet.
Surface Area: The total surface area of a cube is the sum of the areas of all six faces. If the edge length is denoted as 's', the surface area (SA) is given by the formula:

SA = 6s²
Lateral Surface Area: The lateral surface area (LSA) of a cube is the sum of the areas of its four side faces. The formula for the lateral surface area is:

LSA = 4s²

Now that we have a clear understanding of the properties of a cube, let's state the problem we aim to solve:

Problem: If the lateral surface area of a cube is 100 square inches, what is the length of an edge?

We are given the lateral surface area (LSA) of the cube as 100 square inches, and our goal is to find the length of one edge of the cube. This problem requires us to use the formula for the lateral surface area and work backward to find the edge length.

To solve for the edge length, we'll use the formula for the lateral surface area of a cube:

LSA = 4s²

Where LSA is the lateral surface area and 's' is the length of an edge. We are given that LSA = 100 square inches. Let's plug this value into the formula:

100 = 4s²

Now, we need to solve for 's'. First, divide both sides of the equation by 4:

s² = 100 / 4 s² = 25

Next, take the square root of both sides to find the value of 's':

s = √25 s = 5

So, the length of an edge of the cube is 5 inches.

Let's examine the given answer choices:

A. 4 in B. 2√5 in C. 5 in D. 10 in

Based on our calculations, the correct answer is:

C. 5 in

To ensure a comprehensive understanding, let's break down the solution step by step. The problem provides the lateral surface area of a cube, which is the combined area of its four side faces, excluding the top and bottom. The formula for the lateral surface area (LSA) of a cube is LSA = 4s², where 's' represents the length of one edge. We are given that the LSA is 100 square inches.

Set up the equation: We start with the formula LSA = 4s² and substitute the given LSA value:

100 = 4s²
Isolate s²: To find s², we divide both sides of the equation by 4:

100 / 4 = 4s² / 4

25 = s²
Solve for s: To find the length of the edge 's', we take the square root of both sides of the equation:

√25 = √s²

5 = s

Therefore, the length of an edge of the cube is 5 inches.

Understanding why the other options are incorrect can reinforce your grasp of the concept. Let's analyze each incorrect option:

A. 4 in: If the edge length were 4 inches, the lateral surface area would be:

LSA = 4(4²) = 4(16) = 64 square inches

This does not match the given LSA of 100 square inches.

B. 2√5 in: If the edge length were 2√5 inches, the lateral surface area would be:

LSA = 4(2√5)² = 4(4 * 5) = 4(20) = 80 square inches

This also does not match the given LSA of 100 square inches.

D. 10 in: If the edge length were 10 inches, the lateral surface area would be:

LSA = 4(10²) = 4(100) = 400 square inches

This significantly exceeds the given LSA of 100 square inches.

By calculating the LSA for each incorrect edge length, we can clearly see that only the edge length of 5 inches yields the given lateral surface area of 100 square inches.

Understanding the surface area and edge length of cubes has practical applications in various fields. Here are a few examples:

Construction: In construction, calculating the surface area of cubic structures is essential for estimating the materials needed for cladding, painting, or insulation. Knowing the edge length helps in planning the dimensions of modular building components.
Packaging: Cuboidal and cubic boxes are commonly used in packaging. Determining the surface area is crucial for calculating the amount of material required to make the boxes. The edge length is important for fitting products inside the box efficiently.
Engineering: Engineers often work with cubic components in various designs. Calculating the surface area and edge length is necessary for thermal analysis, structural integrity assessments, and optimizing material usage.
Architecture: Architects use cubic forms in building designs for aesthetic and functional purposes. Understanding the dimensions and surface area helps in visualizing the structure and ensuring it meets design requirements.

In this article, we solved the problem of finding the edge length of a cube given its lateral surface area. By understanding the properties of a cube and applying the formula for lateral surface area, we determined that if the lateral surface area of a cube is 100 square inches, the length of an edge is 5 inches. This problem highlights the importance of mastering basic geometric concepts and formulas, which are essential tools for solving more complex problems in mathematics and real-world applications. Remember, the lateral surface area of a cube is a key concept, and the relationship LSA = 4s² allows us to find the edge length effectively. Always double-check your calculations and understand why each step is necessary to arrive at the correct solution. The edge length of a cube is a fundamental property that helps us understand its size and scale, and mastering these concepts is crucial for further studies in geometry and related fields. This exercise reinforces the importance of understanding the properties of a cube, including its faces, edges, and surface area, which are critical in various mathematical and practical contexts. By correctly applying the formula for lateral surface area, we accurately determined the edge length, showcasing the practical application of geometric principles. We hope this detailed explanation enhances your understanding and problem-solving skills in geometry. The ability to calculate the edge length from the lateral surface area is a valuable skill, not just for academic purposes but also for real-world applications in fields like construction, packaging, and engineering. Practice and a clear understanding of the underlying principles are key to mastering such problems. Keep exploring and refining your mathematical skills to tackle more complex challenges with confidence. Remember, geometry is not just about formulas; it’s about visualizing and understanding the shapes and their properties in three-dimensional space.