A cube is a three-dimensional geometric shape bounded by six square faces, all congruent and perpendicular to one another. Each face of a cube is square, and all its edges have equal lengths.

Surface Area of a Cube

The surface area of a cube is the total area of all its faces.

Formula for Surface Area of Cube

The surface area of a cube is the total area of all its faces, and since a cube has six identical square faces, the total surface area of a cube can be calculated using the formula:

Total Surface Area = 6 x (edge/length)^{2}

Where "edge length" is the length of any one side of the cube. This formula calculates the sum of the areas of all six faces of the cube, giving us the total surface area of the cube.

Let's assume each face is a square with an edge length of "a". Area of one face = a^{2} As we know there are a total of six faces.

Therefore, the surface area of a cube with an edge length of "a'' is: Total surface area = a^{2} + a^{2} + a^{2} + a^{2} + a^{2} + a^{2} Total surface area = 6a^{2}

Example: What is the total surface area of the cube if its side is 5 cm?

Solution: Surface area = 6 x (5 x 5) = 6 x 25 = 150 cm²

Lateral Surface Area of a Cube

The lateral surface area of a cube is the sum of the areas of all the faces of the cube that are not its top or bottom faces (i.e., its lateral faces).

Formula for Lateral Surface Area of Cube

The formula for the lateral surface area of a cube is:

Lateral Surface Area = 4 x (side)²

Example: What is the lateral surface area of the cube if its side is 5 cm?

Solution: The lateral surface area would be 4 x (5)² = 4 x 25 = 100 cm^{2}

NOTE: Area is always measured in squares and the unit of area is square units (sq. cm., sq. m. or cm^{2}, m^{2}).

Edge of the Cube

Let the total surface area of a cube be “A”. We know that the total surface area(A) of a cube = 6a^{2} A = 6a^{2} a^{2} = A/6 a = √(A/6)

Here, “a” is the length of the edge of the cube. “A” is the total surface area of a cube.

Diagonal of a Cube

The diagonal of a cube is a line segment that connects two opposite vertices of the cube, passing through the center of the cube. It represents the longest possible distance between any two points within the cube.

Formula for the Diagonal of a Cube:
For a cube with side length "a," the length of the diagonal (d) can be found using the formula:
Diagonal (d) = √(a^{2} + a^{2} + a^{2}) = √(3a^{2}) = √3a

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