Problem 67 - Calculus: One and Several Variables

Question

Express the surface area of a cube as a function of the length of the diagonal of a face.

Step-by-Step Solution

Answer

The surface area of a cube as a function of the length of its face diagonal can be expressed as \(3d^2\).

Step 1 - Understand the relationship between the edge and the diagonal

If we consider one face of the cube as a right triangle by drawing a diagonal over it, the edge length and face diagonal form the two sides and hypotenuse of the triangle, respectively. Using the Pythagorean theorem (\(c^2 = a^2 + b^2\)), where \(c\) is the hypotenuse, \(a\) and \(b\) are sides of the triangle, we can express the edge length (\(a\)) as a function of the face diagonal (\(c\)), since \(a = b\) for a cube.

Step 2 - Solve the equation to find the edge length

We enter our known values into the Pythagorean theorem formula (\(c^2 = a^2 + a^2\)). Simplifying this, we find that \(c^2 = 2a^2\), and therefore, \(a = c / \sqrt{2}\), If we note that the face diagonal (\(c\)) is referred to as (\(d\)), the edge length (\(a\)) can be written as (\(a = d/\sqrt{2}\)). Therefore, the face diagonal is equal to the edge length multiplied by the square root of 2.

Step 3 - Substitute the edge length into the surface area formula

The surface area of a cube can be given by the formula \(6a^2\). Substitute our value of \(a\) (which is \(d/\sqrt{2}\)) from step 2 into the formula. The expression simplifies to \(6(d^2/2)\). Further simplifying, we get the surface area as a function of the face diagonal, which is \(3d^2\).

Key Concepts

Surface Area

The surface area is a key concept when discussing geometry and measures the total area that the surface of a three-dimensional object occupies. For a cube, which consists of six square faces, calculating the surface area involves finding the area of one face and then multiplying it by six.
To understand this in our exercise, we found that once we know the edge length of a cube, calculating the surface area is straightforward using the formula:

• Surface Area = 6a^2

where \(a\) is the edge length. In our exercise, we used the face diagonal instead to express the edge length and then computed the surface area accordingly, showing the beauty and versatility of using geometry in different ways. This allows us to determine the cube's surface area even when directly measuring the edge length is not possible.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It's particularly useful in contexts involving squares or cubes, where diagonal lines are present. The theorem states:

• \( c^2 = a^2 + b^2 \)

Here, \(c\) stands for the hypotenuse, the longest side of the triangle, while \(a\) and \(b\) are the other two sides.
In our exercise, the face of the cube forms a right triangle when a diagonal is drawn. The covergence of this theorem enabled us to express the cube's edge length in terms of the face diagonal:

• \( a = \frac{d}{\sqrt{2}} \)

This expression establishes a critical link, allowing us to leverage the diagonal measurement to find other dimensions, demonstrating the theorem's practical importance in solving real-world problems, even beyond right triangles.

Cube Geometry

Cube geometry is a branch of three-dimensional space that deals specifically with properties and calculations of cubes. A cube has several distinct characteristics that make it special:

• It has six faces, all of which are squares.
• All edges of the cube are of equal length.
• Each vertex joins three edges at right angles.

This geometric structure of cubes allows for interesting applications of mathematical principles, like the Pythagorean theorem, to solve problems. In the exercise, we explored expressing the cube's surface area in terms of the face diagonal by using cube geometry's inherent properties.
Understanding these properties helps build a clear mental picture of spatial relationships and can make tackling complex geometric problems simpler. Mastering cube geometry not only enhances problem-solving skills but also deepens understanding of spatial dimensions, making it a cornerstone topic in calculus and geometry studies.