The Surface Area of a Cube

Oct 13, 2025 by pynkocean.com 4 views

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Introduction

When a cube is cut in half, it results in two identical rectangular prisms. The surface area of the original cube is given as $$C , \text{cm}^2$$, and we are asked to find the surface area of both prisms, expressed as a fraction of the original surface area. In this article, we will delve into the mathematics behind this problem and explore the relationship between the surface area of the cube and the surface area of the resulting prisms.

The Surface Area of a Cube

The surface area of a cube is given by the formula $$6x^2$$, where $$x$$ is the side length of the cube. This formula is derived by calculating the area of each face of the cube and summing them up. Since a cube has six faces, each with an area of $$x^2$$, the total surface area is $$6x^2$$.

Cutting the Cube in Half

When a cube is cut in half, it results in two identical rectangular prisms. Each prism has a length of $$x$$, a width of $$x$$, and a height of $$\frac{x}{2}$$. The surface area of each prism can be calculated by finding the area of each face and summing them up.

The Surface Area of a Rectangular Prism

The surface area of a rectangular prism is given by the formula $$2lw + 2lh + 2wh$$, where $$l$$, $$w$$, and $$h$$ are the length, width, and height of the prism, respectively. In this case, the length and width of each prism are $$x$$, and the height is $$\frac{x}{2}$$.

Calculating the Surface Area of the Prisms

Using the formula for the surface area of a rectangular prism, we can calculate the surface area of each prism:

2x\left(\frac{x}{2}\right) + 2x\left(\frac{x}{2}\right) + 2\left(\frac{x}{2}\right)x

Simplifying the expression, we get:

x^2 + x^2 + x^2

= 3x^2

Comparing the Surface Areas

The surface area of each prism is $$3x^2$$, which is three times the surface area of each face of the original cube. Since the original cube has a surface area of $$6x^2$$, the surface area of both prisms is $$\frac{3}{6}$$ of the original surface area.

Conclusion

In conclusion, when a cube is cut in half, resulting in two identical rectangular prisms, the surface area of both prisms is $$\frac{3}{6}$$ of the original surface area of the cube. This can be expressed as a fraction of the original surface area, $$\frac{M}{N} C$$, where $$M = 3$$ and $$N = 6$$.

Final Answer

The final answer is $$\frac{3}{6}$$, which can be simplified to $$\frac{1}{2}$$.

References

[1] "Surface Area of a Cube" by Math Open Reference
[2] "Surface Area of a Rectangular Prism" by Math Is Fun

Frequently Asked Questions

Q: What is the surface area of a cube?

A: The surface area of a cube is given by the formula $$6x^2$$, where $$x$$ is the side length of the cube.

Q: What happens when a cube is cut in half?

A: When a cube is cut in half, it results in two identical rectangular prisms.

Q: How do you calculate the surface area of a rectangular prism?

A: The surface area of a rectangular prism is given by the formula $$2lw + 2lh + 2wh$$, where $$l$$, $$w$$, and $$h$$ are the length, width, and height of the prism, respectively.

Q: What is the surface area of each prism when a cube is cut in half?

A: The surface area of each prism is $$3x^2$$, which is three times the surface area of each face of the original cube.

Q: How does the surface area of the prisms compare to the surface area of the original cube?

A: The surface area of both prisms is $$\frac{3}{6}$$ of the original surface area of the cube, which can be expressed as a fraction of the original surface area, $$\frac{M}{N} C$$, where $$M = 3$$ and $$N = 6$$.

Q: What is the final answer to the problem?

A: The final answer is $$\frac{3}{6}$$, which can be simplified to $$\frac{1}{2}$$.

Q: What are some related topics to this problem?

A: Some related topics to this problem include surface area of a cube, surface area of a rectangular prism, geometry, and mathematics.

Q: What are some references for this problem?

A: Some references for this problem include "Surface Area of a Cube" by Math Open Reference and "Surface Area of a Rectangular Prism" by Math Is Fun.

Additional Questions and Answers

Q: What if the cube is cut into more than two pieces?

A: If the cube is cut into more than two pieces, the surface area of each piece will be different, and the total surface area will be greater than the surface area of the original cube.

Q: Can the cube be cut into pieces with different dimensions?

A: Yes, the cube can be cut into pieces with different dimensions, but the surface area of each piece will still be related to the surface area of the original cube.

Q: How does the surface area of the prisms change if the cube is cut at different angles?

A: The surface area of the prisms will change if the cube is cut at different angles, but the total surface area will still be related to the surface area of the original cube.

Conclusion

In conclusion, the surface area of a cube and the surface area of the resulting prisms when cut in half are related by a simple ratio. Understanding this relationship can help us solve problems involving surface area and geometry.

Final Answer

The final answer is $$\frac{1}{2}$$.

References

[1] "Surface Area of a Cube" by Math Open Reference
[2] "Surface Area of a Rectangular Prism" by Math Is Fun