Calculating Cube Surface Area A Step-by-Step Guide

Jackson has a cube with each edge measuring 5 centimeters. The crucial task is to determine the total surface area, in square centimeters, that Jackson paints when he covers every face of the cube. This problem delves into the fundamental concept of surface area in three-dimensional geometry, specifically focusing on cubes. To solve this, we will meticulously break down the process, ensuring clarity and a step-by-step understanding. The first critical aspect to recognize is that a cube, by definition, is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Because all faces of a cube are squares and congruent, the area of each face is identical, simplifying the calculation process. To find the total surface area, we first need to calculate the area of one face and then multiply that by the total number of faces, which is six. Given that each edge of the cube is 5 centimeters, each face is a square with sides of 5 centimeters. The area of a square is calculated by squaring the length of one of its sides. Therefore, the area of one face of the cube is 5 cm * 5 cm = 25 square centimeters. Now that we know the area of one face, we can easily find the total surface area by multiplying this value by the number of faces. The cube has six faces, so the total surface area is 25 square centimeters/face * 6 faces = 150 square centimeters. Therefore, the total surface area that Jackson paints is 150 square centimeters. This detailed explanation not only provides the correct answer but also walks through the reasoning, emphasizing the geometric principles at play. Understanding the properties of a cube and the concept of surface area is fundamental in geometry, and this problem serves as an excellent exercise in applying these concepts. Furthermore, this exercise highlights the importance of methodical problem-solving, where breaking down a complex problem into simpler steps leads to an accurate and understandable solution.

Breaking Down the Calculation Step-by-Step

To ensure a comprehensive understanding, let's reiterate the calculation process in a step-by-step manner. This method reinforces the concept and allows for easy application in similar problems. Understanding the fundamentals of geometry, particularly the calculation of surface areas, is crucial for a variety of applications, ranging from everyday tasks such as painting a room to more complex engineering and architectural designs. Breaking down complex calculations into smaller, manageable steps not only makes the process easier to understand but also minimizes the risk of errors. First, we start by identifying the shape in question, which in this case is a cube. A cube is a three-dimensional shape with six equal square faces. Each of these faces contributes to the overall surface area of the cube. The next key piece of information is the length of the cube's edge, which is given as 5 centimeters. This measurement is crucial because it allows us to calculate the area of one of the square faces. The area of a square is found by multiplying the length of one side by itself (side * side). So, for a square with a side of 5 centimeters, the area is 5 cm * 5 cm, which equals 25 square centimeters. Now that we know the area of one face, the next step is to determine the total surface area of the cube. Since a cube has six faces, and each face has the same area, we multiply the area of one face by the total number of faces. This means we multiply 25 square centimeters (area of one face) by 6 (number of faces). The calculation is 25 sq cm * 6 = 150 square centimeters. Therefore, the total surface area of the cube is 150 square centimeters. This step-by-step breakdown not only provides a clear pathway to the solution but also highlights the logical progression required to solve geometric problems. By understanding each step, one can apply this method to similar problems involving different shapes and dimensions. The ability to visualize and deconstruct geometric problems is a valuable skill that enhances problem-solving capabilities in mathematics and beyond.

The Significance of Surface Area in Real-World Applications

Understanding surface area extends far beyond the classroom and plays a vital role in numerous real-world applications. From everyday tasks like wrapping a gift to more complex engineering and architectural projects, the concept of surface area is fundamental. Recognizing its significance can provide a deeper appreciation for the mathematical principles at play in our daily lives. In the context of Jackson's cube, the surface area calculation directly relates to the amount of paint needed to cover all the faces. This is a practical example that can be easily understood: the larger the surface area, the more paint is required. This concept extends to larger-scale projects, such as painting a house or coating a metal structure to prevent corrosion. Calculating the surface area accurately ensures that the right amount of material is purchased, avoiding both waste and the inconvenience of running short. In architecture and construction, surface area is a critical factor in determining the materials needed for building facades, roofing, and interior finishes. Architects and engineers use surface area calculations to estimate costs, plan material procurement, and ensure structural integrity. For instance, the surface area of a building's exterior walls directly influences the amount of cladding material required, while the surface area of the roof affects the amount of roofing material needed. In engineering, the concept of surface area is crucial in designing heat exchangers, which are used in various industries to transfer heat between fluids. The efficiency of a heat exchanger depends largely on its surface area; a larger surface area allows for more heat transfer. Similarly, in the design of solar panels, the surface area of the panel determines the amount of sunlight it can capture and convert into electricity. In the food industry, understanding surface area is essential for packaging design. The surface area of a package influences its cost, the amount of material required, and its ability to protect the contents. For example, calculating the surface area of a cardboard box is crucial for determining the amount of cardboard needed, as well as the printing area for labels and branding. Even in the medical field, surface area calculations have important applications. Body surface area (BSA) is often used to calculate drug dosages and to assess the extent of burns. BSA provides a more accurate measure than weight alone, as it takes into account a person's overall size and shape. These examples illustrate the broad applicability of surface area calculations. By understanding the principles behind these calculations, we can better appreciate the role of mathematics in our daily lives and in various professional fields.

Original: What is the total surface area, in square centimeters, of the cube Jackson paints?

Repaired: What is the total surface area, in square centimeters, of the cube after Jackson paints all its faces?

Calculating Cube Surface Area A Step-by-Step Guide