For Which Of The Following Solids Is The Lateral/curved Surface Area And Total Surface Area The Same?A. Cube B. Cuboid C. Hemisphere D. Sphere

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When dealing with solids, understanding their surface areas is crucial in various mathematical and real-world applications. In this article, we will explore the concept of surface areas of different solids and determine which one has the same lateral/curved surface area and total surface area.

What are Surface Areas?

Surface area is the total area of the surface of a three-dimensional object. It can be calculated by finding the area of each face of the object and adding them together. There are two types of surface areas: total surface area and lateral/curved surface area.

  • Total Surface Area: This is the sum of the areas of all the faces of a solid.
  • Lateral/Curved Surface Area: This is the sum of the areas of all the curved or lateral faces of a solid.

Calculating Surface Areas

To calculate the surface areas of different solids, we need to know their formulas. Here are the formulas for the surface areas of the solids mentioned in the question:

  • Cube: The total surface area of a cube is given by the formula 6a^2, where a is the length of a side. The lateral surface area of a cube is given by the formula 4a^2.
  • Cuboid: The total surface area of a cuboid is given by the formula 2(lb + bh + hl), where l, b, and h are the length, breadth, and height of the cuboid, respectively. The lateral surface area of a cuboid is given by the formula 2(lb + bh + hl) - 2(lb).
  • Hemisphere: The total surface area of a hemisphere is given by the formula 3πr^2, where r is the radius of the hemisphere. The lateral surface area of a hemisphere is given by the formula 2πr^2.
  • Sphere: The total surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere. The lateral surface area of a sphere is given by the formula 0, since a sphere has no lateral faces.

Which Solid has the Same Lateral/Curved Surface Area and Total Surface Area?

From the formulas above, we can see that the only solid that has the same lateral/curved surface area and total surface area is the Sphere. This is because the lateral surface area of a sphere is 0, and the total surface area of a sphere is 4πr^2.

Conclusion

In conclusion, the solid that has the same lateral/curved surface area and total surface area is the Sphere. This is because the lateral surface area of a sphere is 0, and the total surface area of a sphere is 4πr^2. Understanding surface areas of solids is crucial in various mathematical and real-world applications, and this article has provided a comprehensive overview of the concept.

Key Takeaways

  • The total surface area of a solid is the sum of the areas of all its faces.
  • The lateral/curved surface area of a solid is the sum of the areas of all its curved or lateral faces.
  • The only solid that has the same lateral/curved surface area and total surface area is the Sphere.

Frequently Asked Questions

Q: What is the formula for the total surface area of a cube?

A: The formula for the total surface area of a cube is 6a^2, where a is the length of a side.

Q: What is the formula for the lateral surface area of a cuboid?

A: The formula for the lateral surface area of a cuboid is 2(lb + bh + hl) - 2(lb), where l, b, and h are the length, breadth, and height of the cuboid, respectively.

Q: What is the formula for the total surface area of a hemisphere?

A: The formula for the total surface area of a hemisphere is 3πr^2, where r is the radius of the hemisphere.

Q: What is the formula for the lateral surface area of a sphere?

A: The formula for the lateral surface area of a sphere is 0, since a sphere has no lateral faces.

References

In our previous article, we explored the concept of surface areas of different solids and determined which one has the same lateral/curved surface area and total surface area. In this article, we will provide a comprehensive Q&A guide to help you understand surface areas of solids better.

Q: What is the total surface area of a cube?

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

Q: What is the lateral surface area of a cube?

A: The lateral surface area of a cube is given by the formula 4a^2.

Q: What is the total surface area of a cuboid?

A: The total surface area of a cuboid is given by the formula 2(lb + bh + hl), where l, b, and h are the length, breadth, and height of the cuboid, respectively.

Q: What is the lateral surface area of a cuboid?

A: The lateral surface area of a cuboid is given by the formula 2(lb + bh + hl) - 2(lb).

Q: What is the total surface area of a hemisphere?

A: The total surface area of a hemisphere is given by the formula 3πr^2, where r is the radius of the hemisphere.

Q: What is the lateral surface area of a hemisphere?

A: The lateral surface area of a hemisphere is given by the formula 2πr^2.

Q: What is the total surface area of a sphere?

A: The total surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.

Q: What is the lateral surface area of a sphere?

A: The lateral surface area of a sphere is given by the formula 0, since a sphere has no lateral faces.

Q: Which solid has the same lateral/curved surface area and total surface area?

A: The only solid that has the same lateral/curved surface area and total surface area is the Sphere.

Q: Why is the lateral surface area of a sphere 0?

A: The lateral surface area of a sphere is 0 because a sphere has no lateral faces. It is a closed surface, and all its faces are curved.

Q: What is the formula for the surface area of a cone?

A: The formula for the surface area of a cone is given by the formula πr(r + l), where r is the radius of the base and l is the slant height of the cone.

Q: What is the formula for the surface area of a cylinder?

A: The formula for the surface area of a cylinder is given by the formula 2πrh + 2πr^2, where r is the radius of the base and h is the height of the cylinder.

Q: What is the formula for the surface area of a pyramid?

A: The formula for the surface area of a pyramid is given by the formula (1/2)pl + 2b, where p is the perimeter of the base, l is the slant height of the pyramid, and b is the area of the base.

Q: What is the formula for the surface area of a rectangular prism?

A: The formula for 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.

Conclusion

In conclusion, surface areas of solids are an important concept in mathematics and have numerous real-world applications. We hope this Q&A guide has helped you understand surface areas of solids better. If you have any further questions or need clarification on any of the concepts, please feel free to ask.

Key Takeaways

  • The total surface area of a solid is the sum of the areas of all its faces.
  • The lateral/curved surface area of a solid is the sum of the areas of all its curved or lateral faces.
  • The only solid that has the same lateral/curved surface area and total surface area is the Sphere.

Frequently Asked Questions

Q: What is the formula for the total surface area of a cube?

A: The formula for the total surface area of a cube is 6a^2, where a is the length of a side.

Q: What is the formula for the lateral surface area of a cuboid?

A: The formula for the lateral surface area of a cuboid is 2(lb + bh + hl) - 2(lb), where l, b, and h are the length, breadth, and height of the cuboid, respectively.

Q: What is the formula for the total surface area of a hemisphere?

A: The formula for the total surface area of a hemisphere is 3πr^2, where r is the radius of the hemisphere.

Q: What is the formula for the lateral surface area of a sphere?

A: The formula for the lateral surface area of a sphere is 0, since a sphere has no lateral faces.

References