1. Cube
A cube is a 3D solid with all edges equal in length. It has 6 square faces, each of side length \(a\). Because all faces are congruent squares, calculating its surface areas becomes very simple.
Every face has area:
\( a^2 \)
2. Lateral Surface Area of a Cube
The lateral surface area (LSA) includes only the four side faces of the cube. The top and bottom faces are not counted here.
Each side face has area \(a^2\). Since there are 4 such faces:
\( \text{LSA} = 4a^2 \)
2.1. Example
If a cube has side length \(a = 5\,\text{cm}\), then:
\( \text{LSA} = 4a^2 = 4(5^2) = 4 \times 25 = 100\,\text{cm}^2 \)
3. Total Surface Area of a Cube
The Total Surface Area (TSA) includes all six faces of the cube.
Each face has area \(a^2\), so the total is:
\( \text{TSA} = 6a^2 \)
3.1. Why the Formula Makes Sense
A cube has exactly six identical square faces, so multiplying the area of one face by 6 gives the total area. This is the surface area you would need if you wanted to paint or wrap the entire cube.
3.2. Example
If a cube has side length \(a = 8\,\text{cm}\), then:
\( \text{TSA} = 6a^2 = 6(8^2) = 6 \times 64 = 384\,\text{cm}^2 \)
4. Real-Life Applications
Cubes appear often in practical situations:
- Dice
- Ice cubes
- Rubik’s cubes
- Small gift boxes
If a problem asks for the area to paint or wrap the whole cube, use TSA. If only the side area is needed (like sticking a label around the four sides), use LSA.