Formula for Curved Surface Area of a Hemisphere

Oct 13, 2025 by windailys.com 28 views

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Introduction

In this article, we will delve into the world of geometry and explore the concept of a hemisphere's curved surface area. A hemisphere is half of a sphere, and its curved surface area is a crucial aspect of understanding various mathematical concepts. Given the curved surface area of a hemisphere as $$364.5 \pi , \text{cm}^2$$, we will derive a formula to find its radius.

Formula for Curved Surface Area of a Hemisphere

The curved surface area of a hemisphere is given by the formula:

A = 2 \pi r^2

where $$A$$ is the curved surface area and $$r$$ is the radius of the hemisphere.

Derivation of Formula

To derive the formula for the curved surface area of a hemisphere, we can consider it as half of a sphere. The curved surface area of a sphere is given by:

A = 4 \pi r^2

Since a hemisphere is half of a sphere, its curved surface area will be half of the curved surface area of a sphere:

A = \frac{1}{2} \times 4 \pi r^2

A = 2 \pi r^2

Finding the Radius of the Hemisphere

Now that we have the formula for the curved surface area of a hemisphere, we can use it to find the radius of the hemisphere given its curved surface area.

Given the curved surface area of the hemisphere as $$364.5 \pi , \text{cm}^2$$, we can set up the equation:

2 \pi r^2 = 364.5 \pi

To solve for $$r$$, we can divide both sides of the equation by $$2 \pi$$:

r^2 = \frac{364.5 \pi}{2 \pi}

r^2 = 182.25

Taking the square root of both sides of the equation, we get:

r = \sqrt{182.25}

r = 13.5

Conclusion

In this article, we derived the formula for the curved surface area of a hemisphere and used it to find the radius of a hemisphere given its curved surface area. The formula for the curved surface area of a hemisphere is $$A = 2 \pi r^2$$, and we can use it to find the radius of the hemisphere by rearranging the formula and solving for $$r$$. We found that the radius of the hemisphere is $$13.5$$ cm.

Formula Summary

Formula	Description
$$A = 2 \pi r^2$$	Curved surface area of a hemisphere
$$r = \sqrt{\frac{A}{2 \pi}}$$	Radius of a hemisphere

Example Use Case

Suppose we are given the curved surface area of a hemisphere as $$270 \pi , \text{cm}^2$$. We can use the formula for the curved surface area of a hemisphere to find its radius:

2 \pi r^2 = 270 \pi

r^2 = \frac{270 \pi}{2 \pi}

r^2 = 135

r = \sqrt{135}

r = 11.6

Therefore, the radius of the hemisphere is $$11.6$$ cm.

References

[1] "Geometry" by Michael Artin
[2] "Calculus" by Michael Spivak

Note: The references provided are for general information and are not specific to the topic of the article.

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Introduction

In our previous article, we derived the formula for the curved surface area of a hemisphere and used it to find the radius of a hemisphere given its curved surface area. In this article, we will address some common questions related to the topic and provide additional information to help you better understand the concept.

Q&A

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

A: The curved surface area of a hemisphere is given by the formula:

A = 2 \pi r^2

where $$A$$ is the curved surface area and $$r$$ is the radius of the hemisphere.

Q: How do I find the radius of a hemisphere given its curved surface area?

A: To find the radius of a hemisphere given its curved surface area, you can use the formula:

r = \sqrt{\frac{A}{2 \pi}}

where $$A$$ is the curved surface area and $$r$$ is the radius of the hemisphere.

Q: What is the difference between the curved surface area of a hemisphere and a sphere?

A: The curved surface area of a hemisphere is half of the curved surface area of a sphere. The curved surface area of a sphere is given by:

A = 4 \pi r^2

while the curved surface area of a hemisphere is given by:

A = 2 \pi r^2

Q: Can I use the formula for the curved surface area of a hemisphere to find the radius of a sphere?

A: No, the formula for the curved surface area of a hemisphere is specific to hemispheres and cannot be used to find the radius of a sphere.

Q: What are some real-world applications of the concept of the curved surface area of a hemisphere?

A: The concept of the curved surface area of a hemisphere has many real-world applications, including:

Architecture: The curved surface area of a hemisphere is used in the design of domes and other curved structures.
Engineering: The curved surface area of a hemisphere is used in the design of tanks and other containers.
Physics: The curved surface area of a hemisphere is used in the study of gravitational forces and other physical phenomena.

Additional Information

Formula Summary

Formula	Description
$$A = 2 \pi r^2$$	Curved surface area of a hemisphere
$$r = \sqrt{\frac{A}{2 \pi}}$$	Radius of a hemisphere

Example Use Case

Suppose we are given the curved surface area of a hemisphere as $$300 \pi , \text{cm}^2$$. We can use the formula for the curved surface area of a hemisphere to find its radius:

2 \pi r^2 = 300 \pi

r^2 = \frac{300 \pi}{2 \pi}

r^2 = 150

r = \sqrt{150}

r = 12.2

Therefore, the radius of the hemisphere is $$12.2$$ cm.

References

[1] "Geometry" by Michael Artin
[2] "Calculus" by Michael Spivak

Note: The references provided are for general information and are not specific to the topic of the article.

Conclusion

In this article, we addressed some common questions related to the concept of the curved surface area of a hemisphere and provided additional information to help you better understand the concept. We also provided a formula summary and an example use case to help you apply the concept in real-world situations.