Express the edge length of a cube as a function of the cube’s diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length. A function for the edge length of a cube is

Express the edge length of a cube as a function of the cube's diagonal length d - Free 73A

Answer

Cube Dimensions from Diagonal Length – Learnlyfly Geometry Guide

Cube Dimensions from Diagonal Length

In geometry, a cube is a three-dimensional solid with all sides equal in length and all angles at 90 degrees. A unique property of a cube is that its **space diagonal** connects opposite corners through the center of the cube. Using this diagonal, we can derive the cube’s edge length, surface area, and volume.

📏 Understanding the Cube’s Diagonal

The **diagonal (d)** of a cube spans from one vertex to the opposite vertex across the body of the cube. If the edge length of the cube is s, then we can apply the **3D Pythagorean theorem**:

d = √(s² + s² + s²) = √(3s²) = s√3

🔄 Expressing Edge Length in Terms of Diagonal

To express the edge length as a function of the diagonal, solve for s from the formula:

s = d / √3

✅ So, the edge length of the cube in terms of its diagonal d is:

s(d) = d / √3

📐 Surface Area as a Function of Diagonal

The surface area of a cube is given by:

Surface Area = 6s²

Substitute s = d / √3 into the formula:

A(d) = 6 × (d² / 3) = 2d²

✅ So, the surface area of the cube in terms of diagonal d is:

A(d) = 2d²

📦 Volume as a Function of Diagonal

The volume of a cube is given by:

Volume = s³

Again, substitute s = d / √3 into the volume formula:

V(d) = (d / √3)³ = d³ / (3√3)

✅ So, the volume of the cube as a function of diagonal d is:

V(d) = d³ / (3√3)

📊 Summary of Formulas

🔹 Edge length: s(d) = d / √3
🔹 Surface Area: A(d) = 2d²
🔹 Volume: V(d) = d³ / (3√3)

📘 Why This Matters in Geometry

These formulas are not just academic—they’re used in real-world applications such as architecture, 3D modeling, computer graphics, and packaging design. Knowing how to express cube properties in terms of the diagonal allows flexibility in solving spatial problems when direct measurements are unavailable.

📐 Bonus Tip: Use Rationalized Denominators

If your instructor prefers rationalized answers, you can also express the edge length this way:

s = (d√3) / 3

This keeps radicals out of the denominator, which is often preferred in standardized testing and academic formats.

💬 Conclusion

By understanding the spatial relationship between a cube’s edge and its diagonal, we can create useful functions to describe its geometry. Whether you’re a student preparing for exams or an engineer dealing with 3D structures, mastering these transformations makes your problem-solving more versatile and powerful.

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