Understanding the Volume of a Hemisphere

Oct 13, 2025 by magicalceylontours.com 81 views

Introduction

In this article, we will explore the relationship between the volume and surface area of a solid hemisphere. A hemisphere is half of a sphere, and it has a curved surface and a flat base. The volume of a hemisphere is given as $\frac{396}{7} \, \text{cm}^3$ , and we need to find the total surface area of the solid hemisphere.

Understanding the Volume of a Hemisphere

The volume of a hemisphere is given by the formula:

V = \frac{2}{3} \pi r^3

where $r$ is the radius of the hemisphere. In this case, the volume is given as $\frac{396}{7} \, \text{cm}^3$ . We can set up an equation using the formula for the volume of a hemisphere:

\frac{2}{3} \pi r^3 = \frac{396}{7}

Solving for the Radius

To solve for the radius, we can start by isolating $r^3$ :

r^3 = \frac{3 \cdot 396}{2 \pi \cdot 7}

Simplifying the expression, we get:

r^3 = \frac{1188}{14 \pi}

Taking the cube root of both sides, we get:

r = \sqrt[3]{\frac{1188}{14 \pi}}

Calculating the Surface Area

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

A = 2 \pi r^2 + \pi r^2

where $r$ is the radius of the hemisphere. We can substitute the value of $r$ that we found earlier:

A = 2 \pi \left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2 + \pi \left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2

Simplifying the expression, we get:

A = 3 \pi \left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2

Evaluating the Surface Area

To evaluate the surface area, we need to calculate the value of $\left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2$ . We can start by calculating the value of $\sqrt[3]{\frac{1188}{14 \pi}}$ :

\sqrt[3]{\frac{1188}{14 \pi}} = \sqrt[3]{\frac{1188}{14 \cdot 3.14159}}

Evaluating the expression, we get:

\sqrt[3]{\frac{1188}{14 \pi}} \approx 2.5

Now, we can calculate the value of $\left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2$ :

\left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2 \approx 2.5^2

Evaluating the expression, we get:

\left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2 \approx 6.25

Finally, we can calculate the value of the surface area:

A = 3 \pi \left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2 \approx 3 \cdot 3.14159 \cdot 6.25

Evaluating the expression, we get:

A \approx 59.4

However, we need to express the surface area in terms of the given volume. We can start by expressing the surface area in terms of the radius:

A = 3 \pi r^2

Substituting the value of $r$ that we found earlier, we get:

A = 3 \pi \left(\sqrt[3]{\frac{1188}{14 \pi}}\right)^2

Simplifying the expression, we get:

A = 3 \pi \left(\frac{396}{7} \cdot \frac{3}{2 \pi}\right)^{\frac{2}{3}}

Evaluating the expression, we get:

A = 3 \pi \left(\frac{396}{7} \cdot \frac{3}{2 \pi}\right)^{\frac{2}{3}} = \frac{594}{7}

Conclusion

In this article, we explored the relationship between the volume and surface area of a solid hemisphere. We found that the surface area of the hemisphere is given by the formula:

A = 3 \pi r^2

where $r$ is the radius of the hemisphere. We also found that the surface area can be expressed in terms of the given volume:

A = \frac{594}{7}

Therefore, the total surface area of the solid hemisphere is $\frac{594}{7}$ sq. cm.

References

[1] "Hemisphere." Wikipedia, Wikimedia Foundation, 22 Feb. 2023, en.wikipedia.org/wiki/Hemisphere.
[2] "Surface Area of a Hemisphere." Math Open Reference, mathopenref.com/spherehemisphere.html.
[3] "Volume of a Hemisphere." Math Open Reference, mathopenref.com/spherehemisphere.html.
Frequently Asked Questions (FAQs) about the Volume and Surface Area of a Solid Hemisphere =====================================================================================

Q: What is the formula for the volume of a hemisphere?

A: The formula for the volume of a hemisphere is:

V = \frac{2}{3} \pi r^3

where $r$ is the radius of the hemisphere.

Q: How do I find the radius of a hemisphere given its volume?

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

r = \sqrt[3]{\frac{3V}{2 \pi}}

where $V$ is the volume of the hemisphere.

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

A: The formula for the surface area of a hemisphere is:

A = 2 \pi r^2 + \pi r^2

where $r$ is the radius of the hemisphere.

Q: Can I express the surface area of a hemisphere in terms of its volume?

A: Yes, you can express the surface area of a hemisphere in terms of its volume. The surface area can be expressed as:

A = 3 \pi r^2

where $r$ is the radius of the hemisphere.

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

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

A = 3 \pi \left(\frac{3V}{2 \pi}\right)^{\frac{2}{3}}

where $V$ is the volume of the hemisphere.

Q: What is the relationship between the volume and surface area of a hemisphere?

A: The volume and surface area of a hemisphere are related by the formula:

A = 3 \pi r^2

where $r$ is the radius of the hemisphere.

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

A: Yes, you can use the surface area formula to find the radius of a hemisphere. Rearranging the formula, you get:

r = \sqrt{\frac{A}{3 \pi}}

where $A$ is the surface area of the hemisphere.

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

A: The surface area of a hemisphere is significant because it represents the total area of the curved surface and the flat base of the hemisphere. It is an important concept in mathematics and physics, particularly in the study of shapes and volumes.

Q: Can I use the surface area formula to find the volume of a hemisphere?

A: Yes, you can use the surface area formula to find the volume of a hemisphere. Rearranging the formula, you get:

V = \frac{2}{3} \pi r^3

where $r$ is the radius of the hemisphere.

Q: What is the relationship between the surface area and volume of a hemisphere?

A: The surface area and volume of a hemisphere are related by the formula:

A = 3 \pi r^2

where $r$ is the radius of the hemisphere.

Conclusion

In this article, we have answered some of the most frequently asked questions about the volume and surface area of a solid hemisphere. We have explored the formulas for the volume and surface area of a hemisphere, and we have discussed the relationship between the volume and surface area of a hemisphere. We hope that this article has been helpful in clarifying some of the concepts related to the volume and surface area of a hemisphere.