What is the Difference Between Total Surface Area and Lateral Surface Area of a Cube?
The concept of surface area of cube plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It helps us measure the total area that covers the outside surfaces of a three-dimensional cube, a shape commonly seen in boxes, dice, and building blocks.
What Is Surface Area of Cube?
A cube is a three-dimensional shape with six identical square faces. The surface area of a cube means the total area of all these six square faces combined. You'll find this concept applied in areas such as packaging design, geometry word problems, and 3D visualization in maths and science.
Key Formula for Surface Area of Cube
Here’s the standard formula: \( \text{Surface Area of Cube} = 6a^2 \)
where “a” is the length of each edge of the cube. This formula helps you quickly calculate how much material would be needed to cover the entire cube.
Cube Surface Area: Total vs Lateral
Total Surface Area (TSA) includes all 6 faces (TSA = 6a^2).
Lateral Surface Area (LSA) only includes the 4 side faces, ignoring the top and bottom (LSA = 4a^2).
| Area | Formula | Meaning |
|---|---|---|
| Total Surface Area (TSA) | 6a2 | Area of all 6 faces |
| Lateral Surface Area (LSA) | 4a2 | Area of 4 sides only |
Step-by-Step Illustration
- Find the edge length "a" of the cube.
Example: If a = 5 cm, - Square the side: \( a^2 \).
\( 5^2 = 25 \) cm2 - Multiply by 6 for total area:
6 × 25 = 150 cm2 - Write the answer with correct units:
Total Surface Area = 150 cm2
Speed Trick or Vedic Shortcut
Here's a quick hack: If the volume (a3) is given, just find the cube root to get "a", then use 6a2 for surface area.
For example, if volume is 216 cm3:
- Find cube root: \( a = \sqrt[3]{216} = 6 \) cm
- Calculate TSA: 6 × 62 = 6 × 36 = 216 cm2
Vedantu’s live classes often share such tricks to make your calculations faster and easier during tests and competitive exams.
Real-Life Applications of Cube Surface Area
You use surface area of cube when wrapping gifts, painting boxes, boxing packages, or building models. It also comes up in geometry exam questions and computer graphics design.
Try These Yourself
- If a cube’s side is 7 cm, what is its total surface area?
- Can a cube with volume 27 m3 have a surface area of 54 m2?
- Find the lateral surface area for a cube with side 9 mm.
- What happens to the surface area when the side of a cube is doubled?
Frequent Errors and Misunderstandings
- Mixing up the formulas for cube and cuboid.
- Forgetting to square the side (using 6×a instead of 6×a2).
- Confusing total and lateral surface area.
- Writing the answer without square units (cm2, m2).
Relation to Other Concepts
The idea of surface area of cube connects closely with surface area of cuboid and volume of cube. You’ll need these concepts for surface area and volume problems (Class 10) and for understanding square units in calculations.
Classroom Tip
A quick way to remember surface area of cube: “Six faces, all squares, 6a2.” Draw a cube net to visualize all 6 faces clearly—use colored paper squares for better memory. Vedantu’s teachers often show this during maths live classes for visual learners.
We explored surface area of cube—from definition, formula, examples, mistakes, and connections to other maths concepts. Keep practicing with Vedantu to build confidence in problem-solving and score high in your exams!
- Surface Area of Cuboid – Difference and similarities with cube area.
- Square Units: Definition and Uses – Why we use cm2 and m2 in answers.
FAQs on Surface Area of Cube – Formula with Examples and Practice
1. What is the surface area of a cube?
The surface area of a cube is the total area of all six of its square faces. It represents the total area covered by the cube's exterior. The formula for calculating this area is crucial in various geometrical and real-world applications.
2. What is the formula for the surface area of a cube?
The formula for the surface area (SA) of a cube is: SA = 6a², where 'a' represents the length of one side (edge) of the cube. This formula is derived from the fact that a cube has six identical square faces, each with an area of a².
3. How do you calculate the total surface area of a cube if the side is 5 cm?
Given the side length a = 5 cm, we use the formula SA = 6a². Substituting the value, we get: SA = 6 × (5 cm)² = 6 × 25 cm² = 150 cm². Therefore, the total surface area of the cube is 150 square centimeters.
4. What is the difference between total surface area (TSA) and lateral surface area (LSA) of a cube?
The TSA includes the area of all six faces of the cube. The LSA, also known as the curved surface area (CSA), only considers the area of the four side faces, excluding the top and bottom. The formula for LSA is 4a².
5. Why do we use square units (cm², m²) for surface area?
Surface area measures a two-dimensional space. Square units (like cm² or m²) are used because they represent the area occupied by a square with sides of the specified unit length. This is consistent with the units used in measuring the area of any two-dimensional shape.
6. How is the surface area of a cube relevant in real-world applications?
Understanding cube surface area is crucial in various real-world scenarios. For example:
• Packaging: Determining the amount of material needed to manufacture boxes.
• Painting: Calculating the paint required to cover a cubical object.
• Construction: Estimating material needed for cubical structures.
7. What is the surface area of a cube with a volume of 216 cubic centimeters?
First, find the side length. The volume of a cube is a³. Since a³ = 216 cm³, the cube root of 216 is 6 cm (a = 6 cm). Then, use the surface area formula: SA = 6a² = 6 × (6 cm)² = 216 cm².
8. Can I use the surface area formula for a cuboid?
No, the formula 6a² is specific to cubes (all sides equal). Cuboids have different side lengths, requiring a different formula: 2(lb + bh + hl), where l, b, and h are the length, breadth, and height.
9. What happens to the surface area of a cube if its side length doubles?
If the side length doubles (from 'a' to '2a'), the surface area increases fourfold. This is because the new surface area becomes 6(2a)² = 24a², which is four times the original 6a².
10. How can I find the side length of a cube if only the surface area is given?
If the surface area (SA) is known, rearrange the formula: SA = 6a². Solving for 'a', we get: a = √(SA/6). This gives the length of one side of the cube.
11. Explain the concept of a cube net and how it helps visualize surface area.
A cube net is a two-dimensional pattern that can be folded to form a three-dimensional cube. Visualizing a cube net helps understand how the six square faces are interconnected and how their individual areas contribute to the total surface area of the cube.
12. What are some common mistakes students make when calculating the surface area of a cube?
Common mistakes include:
• Forgetting to square the side length before multiplying by 6.
• Using incorrect units (e.g., cubic units instead of square units).
• Confusing surface area with volume.
• Incorrectly applying the formula to shapes other than cubes.
