Surface Area of Cube
A cube is a three-dimensional solid with six identical square faces, twelve equal edges, and eight vertices. It is one of the most fundamental 3D shapes studied in geometry.
The surface area of a cube is the total area of all its six faces. Since each face is a square with the same side length, calculating the surface area is straightforward.
In Class 8 Mensuration, surface area is an important concept. It tells us how much material is needed to cover or wrap a cube-shaped object — whether it is a gift box, a die, or a room.
There are two types of surface area: Total Surface Area (TSA), which includes all six faces, and Lateral Surface Area (LSA), which includes only the four side faces (excluding the top and bottom).
What is Surface Area of Cube?
Definition: The surface area of a cube is the sum of the areas of all its faces.
Key terms:
- Cube — A 3D shape with 6 equal square faces, 12 equal edges, and 8 vertices.
- Edge (a) — The length of one side of any face. All 12 edges of a cube are equal.
- Face — Each flat surface of the cube. A cube has 6 faces, each a square of side a.
- Total Surface Area (TSA) — The area of all 6 faces combined.
- Lateral Surface Area (LSA) — The area of the 4 side faces only (excludes top and bottom).
A cube is a special case of a cuboid where length = breadth = height = a.
Surface Area of Cube Formula
Total Surface Area (TSA) of a Cube:
TSA = 6a²
Where:
- a = length of one edge of the cube
Lateral Surface Area (LSA) of a Cube:
LSA = 4a²
Area of one face:
- Area of one face = a²
Finding edge from TSA:
- a = √(TSA / 6)
Finding edge from LSA:
- a = √(LSA / 4)
Derivation and Proof
Derivation of Total Surface Area:
Step 1: A cube has 6 faces.
Step 2: Each face is a square with side length = a.
Step 3: Area of one square face = a × a = a².
Step 4: Total surface area = 6 × (area of one face) = 6 × a² = 6a².
Derivation of Lateral Surface Area:
Step 1: The lateral surface includes only the 4 vertical (side) faces — it excludes the top and bottom faces.
Step 2: Each side face is a square with area = a².
Step 3: LSA = 4 × a² = 4a².
Alternatively: If you "unfold" the 4 side faces, you get a rectangle with:
- Length = 4a (perimeter of the base)
- Height = a
- Area = 4a × a = 4a²
Relationship:
- TSA = LSA + 2 × (area of base) = 4a² + 2a² = 6a²
Types and Properties
Problems on the surface area of a cube can be categorised as:
1. Finding TSA given the edge:
- Apply TSA = 6a² directly.
2. Finding LSA given the edge:
- Apply LSA = 4a² directly.
3. Finding the edge from TSA or LSA:
- a = √(TSA / 6) or a = √(LSA / 4).
4. Word problems involving painting or wrapping:
- Painting all faces → use TSA.
- Painting only the walls (no floor, no ceiling) → use LSA.
- Wrapping a cube-shaped gift → use TSA.
5. Cost problems:
- Find the area first, then multiply by cost per unit area.
6. Finding the ratio of surface areas:
- If edge is doubled, TSA becomes 4 times. If edge is tripled, TSA becomes 9 times.
Solved Examples
Example 1: Example 1: TSA of a cube
Problem: Find the total surface area of a cube with edge 5 cm.
Solution:
Given:
- a = 5 cm
Using the formula:
- TSA = 6a² = 6 × 5² = 6 × 25 = 150 cm²
Answer: TSA = 150 cm².
Example 2: Example 2: LSA of a cube
Problem: Find the lateral surface area of a cube with edge 8 cm.
Solution:
Given:
- a = 8 cm
Using the formula:
- LSA = 4a² = 4 × 8² = 4 × 64 = 256 cm²
Answer: LSA = 256 cm².
Example 3: Example 3: Finding edge from TSA
Problem: The total surface area of a cube is 294 cm². Find the edge length.
Solution:
Given:
- TSA = 294 cm²
Using the formula:
- 6a² = 294
- a² = 294 / 6 = 49
- a = √49 = 7 cm
Answer: The edge length is 7 cm.
Example 4: Example 4: Cost of painting a cube
Problem: A cube-shaped water tank has an edge of 3 m. If painting costs Rs. 50 per m², find the cost of painting the outer surface.
Solution:
Step 1: TSA = 6 × 3² = 6 × 9 = 54 m²
Step 2: Cost = 54 × 50 = Rs. 2,700
Answer: The cost of painting is Rs. 2,700.
Example 5: Example 5: Painting only the walls
Problem: A cube-shaped room has side 4 m. Find the area of the four walls.
Solution:
Given:
- a = 4 m
Using LSA (four walls only):
- LSA = 4a² = 4 × 4² = 4 × 16 = 64 m²
Answer: The area of the four walls is 64 m².
Example 6: Example 6: Wrapping a gift box
Problem: A gift box is cube-shaped with edge 15 cm. How much wrapping paper is needed? (Assume no overlap.)
Solution:
Given:
- a = 15 cm
Wrapping paper needed = TSA:
- TSA = 6 × 15² = 6 × 225 = 1,350 cm²
Answer: 1,350 cm² of wrapping paper is needed.
Example 7: Example 7: Edge doubled — effect on TSA
Problem: If the edge of a cube is doubled, by how much does the total surface area increase?
Solution:
Original: TSA₁ = 6a²
New edge = 2a: TSA₂ = 6(2a)² = 6 × 4a² = 24a²
Ratio: TSA₂ / TSA₁ = 24a² / 6a² = 4
Answer: The TSA becomes 4 times the original. It increases by 3 times (300% increase).
Example 8: Example 8: Finding edge from LSA
Problem: The lateral surface area of a cube is 100 cm². Find the edge and the total surface area.
Solution:
Step 1: Find the edge:
- LSA = 4a²
- 100 = 4a²
- a² = 25
- a = 5 cm
Step 2: TSA = 6a² = 6 × 25 = 150 cm²
Answer: Edge = 5 cm, TSA = 150 cm².
Example 9: Example 9: Open-top cube-shaped tank
Problem: A cube-shaped water tank (open at the top) has edge 2 m. Find the surface area to be painted.
Solution:
Given:
- a = 2 m, open at top (5 faces to paint)
Surface area:
- = TSA − area of one face
- = 6a² − a²
- = 5a²
- = 5 × 2² = 5 × 4 = 20 m²
Answer: The area to be painted is 20 m².
Example 10: Example 10: Comparing surface areas
Problem: Cube A has edge 6 cm and Cube B has edge 10 cm. Find the difference in their total surface areas.
Solution:
TSA of Cube A:
- = 6 × 6² = 6 × 36 = 216 cm²
TSA of Cube B:
- = 6 × 10² = 6 × 100 = 600 cm²
Difference:
- = 600 − 216 = 384 cm²
Answer: The difference in TSA is 384 cm².
Real-World Applications
Packaging and Wrapping:
- To wrap a cube-shaped gift box or package, you need to know the total surface area to cut the right amount of wrapping paper or cardboard.
Painting and Coating:
- When painting a cube-shaped room (walls, ceiling, floor), surface area determines how much paint is needed. Painters calculate LSA for walls and add floor/ceiling as needed.
Manufacturing:
- Manufacturing cube-shaped containers, dice, or building blocks requires surface area calculations for material cost estimation.
Construction:
- Cube-shaped water tanks and storage units need surface area calculations for waterproofing, insulation, and material estimation.
Cost Estimation:
- Knowing the surface area helps calculate the cost of materials — tiling, carpeting, laminating, or polishing a cube-shaped object.
Key Points to Remember
- A cube has 6 equal square faces, 12 equal edges, and 8 vertices.
- Total Surface Area (TSA) = 6a², where a is the edge length.
- Lateral Surface Area (LSA) = 4a² (four side faces only).
- Area of one face = a².
- TSA = LSA + 2a² (LSA + top + bottom).
- To find the edge from TSA: a = √(TSA / 6).
- To find the edge from LSA: a = √(LSA / 4).
- If the edge is doubled, TSA becomes 4 times.
- If the edge is tripled, TSA becomes 9 times.
- A cube is a special cuboid where l = b = h = a.
Practice Problems
- Find the TSA of a cube with edge 12 cm.
- Find the LSA of a cube with edge 9 cm.
- The TSA of a cube is 486 cm². Find the edge length.
- A cube-shaped box has edge 20 cm. Find the amount of cardboard needed to make it (in cm²).
- A cube-shaped room has side 5 m. If painting costs Rs. 25 per m², find the cost of painting all four walls.
- The edge of a cube is tripled. What happens to its total surface area?
- An open-top cube-shaped tank has edge 1.5 m. Find the area of the metal sheet used to make it.
- Two cubes of edge 3 cm each are joined end to end. Find the surface area of the resulting cuboid.
Frequently Asked Questions
Q1. What is the total surface area of a cube?
The total surface area (TSA) of a cube with edge a is 6a². It includes the area of all 6 square faces.
Q2. What is the difference between TSA and LSA of a cube?
TSA (6a²) includes all 6 faces. LSA (4a²) includes only the 4 side faces, excluding the top and bottom. TSA = LSA + 2a².
Q3. How do you find the edge of a cube if TSA is given?
Use a = √(TSA / 6). For example, if TSA = 150 cm², then a = √(150 / 6) = √25 = 5 cm.
Q4. Why does TSA become 4 times when the edge is doubled?
TSA = 6a². If edge becomes 2a, new TSA = 6(2a)² = 6 × 4a² = 24a². Ratio = 24a² / 6a² = 4. The surface area scales as the square of the scaling factor.
Q5. Is a cube a special type of cuboid?
Yes. A cube is a cuboid where length = breadth = height = a. The cuboid formulas TSA = 2(lb + bh + hl) and LSA = 2h(l + b) reduce to 6a² and 4a² respectively when l = b = h = a.
Q6. When do we use LSA instead of TSA?
Use LSA when only the side faces are involved — for example, painting the four walls of a cube-shaped room (not the floor or ceiling), or wrapping a label around a cube-shaped container.
Q7. What is the surface area of a cube with edge 1 cm?
TSA = 6 × 1² = 6 cm². This cube is often called a 'unit cube'.
Q8. How does surface area relate to volume for a cube?
TSA = 6a² and Volume = a³. As the cube gets larger, volume grows faster than surface area. For edge a, Volume/TSA = a/6.
Q9. What is the surface area of a hollow cube?
A hollow cube (like a box without a lid) has 5 faces. Its surface area = 5a². If both top and bottom are missing, the area = 4a² (same as LSA).
Q10. How many faces of a Rubik's cube are visible from outside?
A standard Rubik's cube has 6 visible faces. Its surface area equals the TSA of the cube = 6a², where a is the edge length.
