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Chapter 12: Problem 1

What is the surface area of a cube with sides of \(3.1 \mathrm{~cm}\) ?

Short Answer

Expert verified
Answer: The surface area of the cube is 57.66 cm虏.

Step by step solution

01

Find the area of one face of the cube

To find the area of one face of the cube, you need to square the side length. In this case, the side length is \(3.1 \mathrm{~cm}\). So, the area of one face is: \((3.1 \mathrm{~cm})^2 = 9.61 \mathrm{cm^2}\)
02

Find the total surface area of the cube

Since a cube has 6 identical faces, to find the total surface area, you need to multiply the area of one face by 6. So, the total surface area of the cube is: \(6 \times 9.61 \mathrm{cm^2} = 57.66 \mathrm{cm^2}\) The surface area of the cube with sides of \(3.1 \mathrm{~cm}\) is \(57.66 \mathrm{cm^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry
Geometry is a branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. Understanding geometry is crucial to visualize and solve problems like determining the surface area of a three-dimensional object.

In the case of a cube, which is a regular, three-dimensional object with six square faces, all sides are equal in length. Each angle is a right angle, and each face is a square. Therefore, understanding the properties of squares and how they contribute to the cube's overall geometry is essential in solving problems related to cubes.
Square
A square is a four-sided polygon called a quadrilateral. It has all four sides of equal length and each of the four angles is a right angle (90 degrees). The area of a square is found by multiplying the length of one side by itself, which is squaring the side length.

When it comes to calculating the area for one face of a cube, you're essentially calculating the area of a square since all faces of a cube are squares. The relationship between a cube and its square faces is fundamental in geometry as it leads to a better understanding of the cube's surface area calculation.
Math Problem Solving
Math problem solving involves a set process of understanding the problem, devising a plan, carrying out the plan, and then evaluating the results. For geometric problems, visualization and systematic approach are key.

In the context of the exercise, solving for the surface area of a cube entails recognizing that there are six equal square faces and the area of these faces must be calculated first. The approach systematically follows a two-step process: first find the area of one face (a square), then multiply by the number of faces (six for a cube).
Area Calculation
Area calculation in geometry often requires knowing the formula for the shape you are dealing with. For squares, the formula is side length squared. Since the cube's surface area is the total area of all six squares, you multiply the area of one square by six.

The exercise given demonstrates this by squaring the side length of the cube to get the area of one face, and then multiplying this area by six to get the cube's total surface area. This calculation is essential for understanding how to approach and solve area-related questions in mathematics.

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