Surface Area of Cube

Introduction

In geometry,  surface area  is an essential concept that allows us to determine the amount of area covered on the exterior of a cube. Whether wrapping a gift box or painting dice, determining the surface area of a cube is an important everyday skill. The formula for the surface area of a cube provides a simple means of calculating the area precisely.

A cube has six equal square faces, so finding its surface area is simple once you know the formula. By multiplying the area of one face by six, you can calculate the total surface area of a cube quickly and accurately. This guide will explain the concept, formula, derivation, examples, and real-life applications of the surface area of a cube.

 

Table of Contents

• What is a Cube?
• Understanding Surface Area
• Surface Area of Cube
• Derivation of the Surface Area of a Cube Formula
• Total Surface Area of Cube (TSA)
• TSA of Cube Formula with Units
• Real-life Examples
• Solved Examples
• Practice Problems
• Conclusion
• FAQs On the Surface Area of a Cube

 

What is a Cube?

A cube is a 3D solid shape, has 6 square faces, 12 edges, and 8 corners. All sides are the same length, and each face meets another at a right angle. In simple words, a cube looks like a box or dice, where each side is the same.

Characteristics of a Cube:

• 6 faces (all squares)

• 12 equal edges

• 8 vertices

• All interior angles = 90°

• All faces are congruent squares

 

Understanding Surface Area

Before learning how to find the surface area of a cube, it’s important to know what surface area means.

Surface area is the total area that covers the outer surface of a three-dimensional (3D) object. In other words, it is the amount of space on the exterior of a solid shape. For a cube, which has six equal square faces, the surface area is the sum of the areas of all its faces. So, to find the surface area of a cube, you first calculate the area of one face and then multiply it by 6.

Knowing the surface area is useful in real life, for example, when wrapping a box, painting a cube-shaped object, or designing cube-like structures.

 

Surface Area of Cube

The surface area is the total area covered by all the outer faces of a cube. Since all the faces are squares and each side has the same length, finding the surface area is straightforward.

To calculate the surface area, we first find the area of one face. If the length of a side is a, then the area of one face is:

Area of one face=a×a=a²

A cube has six faces, so the total surface area is the sum of the areas of all six faces:

Total Surface Area=6×a²

This is the formula for surface area of a cube, which is applied in every general cube calculation.

 

Surface Area of Cube Formula

Let's define it officially:

Surface Area of Cube = 6a²

Where,

• a = side length of one edge of the cube

• 6 = number of square faces

• a² = area of a single square face

The surface area of cube formula works universally for any cube irrespective of the size.

 

Derivation of the Surface Area of Cube Formula

There are six square faces on every cube.

 The area of each face is a².

 Multiply that by 6:

Total Surface Area of Cube = 6 × a²

Therefore,

 The formula of tsa of a cube is:

 TSA = 6a²

The formula of tsa of cube can easily and precisely be computed in exams at school and even in practical life.

 

Total Surface Area of Cube (TSA)

Total surface area of cube is the sum of the areas of all six square faces of the cube. It is the complete outer covering of the cube.

Example:

What is the total surface area of cube having an edge length of 4 cm?

Apply the surface area of cube formula:

TSA = 6a² = 6 × 4² = 6 × 16 = 96 cm²

Thus, the total surface area of cube is 96 cm².

 

TSA of Cube Formula with Units

When using the tsa of cube formula, always mention proper units:

• Edge in cm → TSA in cm²
• Edge in m → TSA in m²

Example:

 Edge = 5 m

 TSA = 6 × 5² = 6 × 25 = 150 m²

The surface area of the cube is 150 m² using the tsa of cube formula.

 

Real-life Examples of Cube Surface Area

• Gift Box: Gift wrapping a cube-shaped gift requires finding the total surface area of the cube to determine how much wrapping paper would be required.

• Dice: The Paint required to paint each side of a dice is based on the surface area of the cube.

• Shipping: In calculating package material for cube boxes, the formula for tsa of a cube is applied.

 

Solved Examples

Solved Example 1: 

Question:
Find the surface area of a cube whose edge length is 5 cm.

Solution:
Use the surface area of cube formula:
Surface Area of Cube = 6a²
a = 5 cm
= 6 × (5)²
= 6 × 25
= 150 cm²

Answer: The total surface area of the cube is 150 cm².

 

Solved Example 2: 

Question:
Using the tsa of cube formula, calculate the surface area of cube with edge = 12 cm.

Solution:
TSA of Cube Formula:
TSA = 6a²
= 6 × (12)²
= 6 × 144
= 864 cm²

Answer: The surface area of cube is 864 cm².

 

Solved Example 3: 

Question:
If the surface area of a cube is 294 cm², find the length of one edge.

Solution:
Use the surface area of a cube formula:
6a² = 294
a² = 294 ÷ 6 = 49
a = √49 = 7 cm

Answer: Edge length of the cube is 7 cm.

 

Solved Example 4

Question:
A cube-shaped gift box has a side of 8 cm. How much wrapping paper is needed to cover it completely?

Solution:
Total surface area of cube = 6a²
= 6 × 8²
= 6 × 64
= 384 cm²

Answer: You need 384 cm² of wrapping paper.

 

Solved Example 5:

Question:
A cube has an edge of 2 meters. What is its surface area?

Solution:
TSA = 6a²
= 6 × 2²
= 6 × 4
= 24 m²

Answer: The surface area of cube is 24 m².

Practice Problems

1. Calculate the surface area of cube with edge = 7 cm.
2. A cube with edge = 10 m. What is its total surface area of cube?
3. If the surface area of a cube is 486 cm², determine the length of one side.
4. By applying the tsa of cube formula, determine the TSA for a cube with an edge of 3.5 m.

 

Conclusion

The surface area of a cube is an essential geometric principle with a lot of real-world applications in packaging, design, and mathematics. With the surface area of cube formula under your belt, you can easily determine how much area is occupied by the outer surface of a cube in a split second. Whether you name it the tsa of cube formula or use the term total surface area of cube, the process is the same.

Frequently Asked Questions On Surface Area of Cube

1. What is CSA TSA and volume of cube?

Answer:

• CSA (Curved Surface Area) does not apply to cubes as they have flat faces.
• TSA (Total Surface Area) of a cube is the sum of areas of all 6 square faces, calculated as TSA = 6a².
• Volume of a cube is the amount of space it occupies, calculated by Volume = a³, where a is the side length.
  
  

2. How to calculate the surface area?

Answer: To calculate the surface area of a cube, use the formula:
Surface Area = 6 × side²
For example, if the side is 5 cm, then Surface Area = 6 × 5² = 6 × 25 = 150 cm².

3. What is the surface area of a 3x3 Rubik's cube?

Answer: A standard 3×3 Rubik's Cube has 6 faces, and each face is a 3×3 square.
If each small square is 1 cm², then:
Area of 1 face = 3 × 3 = 9 cm²
Total surface area = 6 × 9 = 54 cm²

4. Is surface area cm² or cm³?

Answer: Surface area is always measured in square units, so the correct unit is cm² (square centimeters). cm³ (cubic centimeters) is used for volume, not surface area.

Discover how to calculate the surface area of a cube with help from Orchids The International School.