Write The Formula That Shows The Dependence Of The Surface Area { S $}$ Of A Cube On The Length { A $}$ Of An Edge.Answer: ${ S = 6a^2 }$

Introduction

In geometry, a cube is a three-dimensional solid object with six square faces, twelve straight edges, and eight vertices. The surface area of a cube is the total area of its six faces, while the length of an edge is the distance between two adjacent vertices. In this article, we will derive the formula that shows the dependence of the surface area of a cube on the length of an edge.

The Formula for the Surface Area of a Cube

The surface area of a cube can be calculated by finding the area of one face and multiplying it by 6, since all faces are identical. The area of one face is given by the formula:

$$A = a^2$$

where $a$ is the length of an edge. Since there are six faces, the total surface area of the cube is:

$$S = 6A = 6a^2$$

Derivation of the Formula

To derive the formula for the surface area of a cube, we need to consider the geometry of the cube. Each face of the cube is a square with side length $a$. The area of a square is given by the formula:

$$A = s^2$$

where $s$ is the side length. In this case, the side length is $a$, so the area of one face is:

$$A = a^2$$

Since there are six faces, the total surface area of the cube is:

$$S = 6A = 6a^2$$

Properties of the Formula

The formula for the surface area of a cube has several important properties:

• Linearity: The formula is linear in the length of an edge, meaning that the surface area increases quadratically with the length of an edge.
• Homogeneity: The formula is homogeneous of degree 2, meaning that the surface area is proportional to the square of the length of an edge.
• Dimensionality: The formula is dimensionless, meaning that the surface area is independent of the units of measurement.

Applications of the Formula

The formula for the surface area of a cube has several important applications in mathematics and physics:

• Geometry: The formula is used to calculate the surface area of a cube in geometry problems.
• Physics: The formula is used to calculate the surface area of a cube in problems involving heat transfer, fluid dynamics, and electromagnetism.
• Engineering: The formula is used to calculate the surface area of a cube in problems involving structural analysis, mechanical engineering, and materials science.

Conclusion

In conclusion, the formula for the surface area of a cube is given by:

$$S = 6a^2$$

This formula shows the dependence of the surface area of a cube on the length of an edge. The formula has several important properties, including linearity, homogeneity, and dimensionality. The formula has several important applications in mathematics and physics, including geometry, physics, and engineering.

References

• [1] "Geometry" by Michael Spivak
• [2] "Physics for Scientists and Engineers" by Paul A. Tipler
• [3] "Engineering Mechanics" by Russell C. Hibbeler

Further Reading

For further reading on the topic of the surface area of a cube, we recommend the following resources:

• [1] "The Surface Area of a Cube" by Math Open Reference
• [2] "Cube Surface Area" by Wolfram MathWorld
• [3] "Surface Area of a Cube" by Khan Academy
  Q&A: The Surface Area of a Cube =====================================

Introduction

In our previous article, we derived the formula for the surface area of a cube, which is given by:

$$S = 6a^2$$

where $a$ is the length of an edge. In this article, we will answer some frequently asked questions about the surface area of a cube.

Q: What is the surface area of a cube with an edge length of 5 units?

A: To find the surface area of a cube with an edge length of 5 units, we can plug in the value of $a$ into the formula:

$$S = 6a^2$$

$$S = 6(5)^2$$

$$S = 6(25)$$

$$S = 150$$

So, the surface area of a cube with an edge length of 5 units is 150 square units.

Q: How does the surface area of a cube change when the edge length is doubled?

A: To find out how the surface area of a cube changes when the edge length is doubled, we can plug in the value of $a$ into the formula:

$$S = 6a^2$$

When the edge length is doubled, the new value of $a$ is $2a$. Plugging this into the formula, we get:

$$S = 6(2a)^2$$

$$S = 6(4a^2)$$

$$S = 24a^2$$

So, the surface area of a cube with an edge length of $2a$ is 24 times the surface area of a cube with an edge length of $a$.

Q: What is the surface area of a cube with an edge length of 10 units and a height of 5 units?

A: To find the surface area of a cube with an edge length of 10 units and a height of 5 units, we need to find the surface area of the cube first. The formula for the surface area of a cube is:

$$S = 6a^2$$

Plugging in the value of $a$, we get:

$$S = 6(10)^2$$

$$S = 6(100)$$

$$S = 600$$

So, the surface area of the cube is 600 square units. However, the question asks for the surface area of a cube with a height of 5 units, which is not a standard cube. We need to find the surface area of a rectangular prism with a height of 5 units and a base area of 100 square units.

The surface area of a rectangular prism is given by:

$$S = 2lw + 2lh + 2wh$$

where $l$, $w$, and $h$ are the length, width, and height of the prism, respectively. In this case, the length and width are both 10 units, and the height is 5 units. Plugging these values into the formula, we get:

$$S = 2(10)(10) + 2(10)(5) + 2(10)(5)$$

$$S = 200 + 100 + 100$$

$$S = 400$$

So, the surface area of a rectangular prism with a height of 5 units and a base area of 100 square units is 400 square units.

Q: How does the surface area of a cube change when the edge length is increased by 10%?

A: To find out how the surface area of a cube changes when the edge length is increased by 10%, we can plug in the value of $a$ into the formula:

$$S = 6a^2$$

When the edge length is increased by 10%, the new value of $a$ is $1.1a$. Plugging this into the formula, we get:

$$S = 6(1.1a)^2$$

$$S = 6(1.21a^2)$$

$$S = 7.26a^2$$

So, the surface area of a cube with an edge length of $1.1a$ is 7.26 times the surface area of a cube with an edge length of $a$.

Conclusion

In conclusion, we have answered some frequently asked questions about the surface area of a cube. We have shown how to calculate the surface area of a cube with different edge lengths and heights, and how the surface area changes when the edge length is doubled, increased by 10%, or changed in other ways.