Problem 25 Find the surface area of a cube ... [FREE SOLUTION] (2024)

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A cube is a three-dimensional shape with six identical square faces. Every face has the same side length, known as the edge.

Cubes belong to a category of shapes known as regular polyhedra, where all faces are not only congruent but also perfectly symmetrical. This consistency in shape makes calculations like volume and surface area straightforward.

The surface area of a cube is essentially the total area of all six faces.
Since each face is a square, you can calculate the area of one face and then multiply by six.

Here's the formula you need:

• Each face area: \(a^2\)
• Total surface area: \(6a^2\)

Where \(a\) is the edge length of the cube.

Understanding and using the correct formula is key.
The surface area formula for a cube is: \( \text{Surface Area} = 6a^2 \)
This formula tells you to square the edge length \(a\), multiply by 6, and you get the surface area in square units.

For example:

• With edge length \(x\), surface area is \(6x^2\)
• With edge length \(2x\), surface area becomes \(6(2x)^2 = 24x^2\)
• With edge length \(3x\), surface area is \(6(3x)^2 = 54x^2\)

The surface area of a cube changes significantly when the edge length changes.
It's not a simple doubling or tripling relationship but rather a squaring and multiplying by six.
For instance:

• If the edge length is doubled (from \(x\) to \(2x\)), the surface area multiplies by 4, giving you \(24x^2 = 6 \times (2x)^2\).
• If the edge length is tripled (from \(x\) to \(3x\)), the surface area multiplies by 9, resulting in \(54x^2 = 6 \times (3x)^2\).

So, a small change in edge length leads to a big change in surface area.