Geometry Puzzle – Challenge 75

Challenge:

If the total surface area of a cube is 3a square feet and the volume of the cube is 2a cubic feet, then what is the total length, in feet, of all the cube’s edges?

A- 12

B- 24

C- 48

D- 64

E- 96

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A cube has 12 edges. Let x be the edge of the cube. So:
Surface area of the cube = 6 (one side)\(^2 = 6x^2 =3a\)
\(x^2 = \frac{1}{2} a → x = \sqrt{\frac{1}{2 }a} = (\frac{1}{2} a)^{\frac{1}{2}}\)
Volume of the cube = (one side)\(^3 = x^3 =2a → x = \sqrt[3]{2a} = (2a)^{\frac{1}{3}}\)
Solve for a for both equations:
\((\frac{1}{2} a)^{\frac{1}{2}} = (2a)^{\frac{1}{3}}\)
Both sides to the power of 6:
\((\frac{1}{2} a)^3 = (2a)^2 → \frac{1}{8} a^3 = 4a^2
→ \frac{1}{8} a = 4 → a = 32\)
Therefore, the volume of the cube is 64 cubic feet and one side is 4.
The total length of all the cube’s edges is 48 feet. (4 × 12 = 48)

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Frequently Asked Questions

How many edges does a cube have?

12 edges — three sets of 4 parallel edges, one set per pair of opposite faces.

Why is surface area 6x^2?

A cube has 6 congruent square faces, each with side x. Each face area is x^2, so the total surface area is 6x^2.

Why is volume x^3?

Length times width times height = x times x times x = x^3.

How do I solve the two equations?

From 6x^2 = 3a, get a = 2x^2. Substitute into x^3 = 2a: x^3 = 2(2x^2) = 4x^2. Divide both sides by x^2: x = 4.

Why can I divide by x^2?

Because x is the edge length of a real cube, so x is positive (nonzero). Dividing both sides by x^2 is valid.

What is a (the parameter)?

From a = 2x^2 with x = 4: a = 2(16) = 32. So the cube has surface area 3(32) = 96 and volume 2(32) = 64. Edge = 4, total edges = 48.

How can I check the answer?

Verify: edge 4 means surface area = 6(16) = 96 = 3(32) ✓. Volume = 64 = 2(32) ✓. Total edges = 12(4) = 48.

Is x = 0 also a mathematical solution?

Yes — x^3 = 4x^2 has solutions x = 0 and x = 4. But x = 0 gives a degenerate “cube” with no size, which is not a real cube.

What if surface area and volume were 6a and a?

Set 6x^2 = 6a (so a = x^2) and x^3 = a = x^2. Divide by x^2: x = 1. Total edges = 12.

Why is this kind of problem valuable?

It connects multiple geometric measures (surface area, volume, edge length) via a shared parameter. The translation skill — relating expressions for different quantities — is the heart of algebra.

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If your student enjoys puzzles like this, Geometry for Beginners works the same relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebraic foundations gently.