Volume of a Hemisphere Inside a Rectangular Prism

Oct 13, 2025 by pinkandtrash.com 28 views

Introduction

In this article, we will explore the concept of volumes of three-dimensional shapes, specifically a hemisphere inside a rectangular prism and a sphere. We will derive the formulas for the volumes of these shapes and apply them to solve problems.

Volume of a Hemisphere Inside a Rectangular Prism

A rectangular prism has a square base with an edge length of 4 cm. Inside the rectangular prism is a hemisphere. To find the volume of the hemisphere, we need to know its radius. Since the hemisphere is inscribed in the rectangular prism, its diameter is equal to the edge length of the square base, which is 4 cm. Therefore, the radius of the hemisphere is half of the diameter, which is 2 cm.

The formula for the volume of a hemisphere is:

V = (2/3)πr^3

where V is the volume and r is the radius of the hemisphere.

Substituting the value of r = 2 cm into the formula, we get:

V = (2/3)π(2)^3 V = (2/3)π(8) V = (16/3)π

Therefore, the volume of the hemisphere in terms of π is (16/3)π cubic centimeters.

Volume of a Sphere

A sphere is a three-dimensional shape that is perfectly round. The formula for the volume of a sphere is:

V = (4/3)πr^3

where V is the volume and r is the radius of the sphere.

Given that the diameter of the sphere is 28 cm, we can find the radius by dividing the diameter by 2:

r = 28/2 r = 14 cm

Substituting the value of r = 14 cm into the formula, we get:

V = (4/3)π(14)^3 V = (4/3)π(2744) V = (10976/3)π

Therefore, the volume of the sphere in terms of π is (10976/3)π cubic centimeters.

Discussion

In this article, we have derived the formulas for the volumes of a hemisphere and a sphere. We have also applied these formulas to solve problems involving a hemisphere inside a rectangular prism and a sphere with a given diameter.

The volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius of the hemisphere. The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.

We have also seen that the volume of a hemisphere inside a rectangular prism is (16/3)π cubic centimeters, and the volume of a sphere with a diameter of 28 cm is (10976/3)π cubic centimeters.

Conclusion

In conclusion, the volume of a hemisphere inside a rectangular prism and a sphere can be calculated using the formulas V = (2/3)πr^3 and V = (4/3)πr^3, respectively. We have applied these formulas to solve problems involving a hemisphere inside a rectangular prism and a sphere with a given diameter.

References

[1] "Mathematics for Dummies" by Mary Jane Sterling
[2] "Calculus for Dummies" by Mark Ryan
[3] "Geometry for Dummies" by Mark Ryan

Mathematical Formulas

Volume of a hemisphere: V = (2/3)πr^3
Volume of a sphere: V = (4/3)πr^3

Mathematical Symbols

π (pi): a mathematical constant representing the ratio of a circle's circumference to its diameter
r: the radius of a hemisphere or a sphere
V: the volume of a hemisphere or a sphere
Volume of a Hemisphere Inside a Rectangular Prism and a Sphere: Q&A ====================================================================

Introduction

In our previous article, we explored the concept of volumes of three-dimensional shapes, specifically a hemisphere inside a rectangular prism and a sphere. We derived the formulas for the volumes of these shapes and applied them to solve problems. In this article, we will answer some frequently asked questions (FAQs) related to the volume of a hemisphere inside a rectangular prism and a sphere.

Q&A

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

A: The formula for the volume of a hemisphere is V = (2/3)πr^3, where r is the radius of the hemisphere.

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

A: The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere.

Q: How do I find the radius of a hemisphere or a sphere?

A: To find the radius of a hemisphere or a sphere, you need to know its diameter. The radius is half of the diameter.

Q: What is the volume of a hemisphere with a radius of 3 cm?

A: To find the volume of a hemisphere with a radius of 3 cm, you can use the formula V = (2/3)πr^3. Substituting the value of r = 3 cm into the formula, we get:

V = (2/3)π(3)^3 V = (2/3)π(27) V = (54/3)π V = 18π

Therefore, the volume of the hemisphere is 18π cubic centimeters.

Q: What is the volume of a sphere with a radius of 5 cm?

A: To find the volume of a sphere with a radius of 5 cm, you can use the formula V = (4/3)πr^3. Substituting the value of r = 5 cm into the formula, we get:

V = (4/3)π(5)^3 V = (4/3)π(125) V = (500/3)π V = (500/3)π

Therefore, the volume of the sphere is (500/3)π cubic centimeters.

Q: Can I use the formula for the volume of a sphere to find the volume of a hemisphere?

A: No, you cannot use the formula for the volume of a sphere to find the volume of a hemisphere. The formula for the volume of a sphere is V = (4/3)πr^3, which is different from the formula for the volume of a hemisphere, V = (2/3)πr^3.

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

A: The volume of a hemisphere is half of the volume of a sphere with the same radius. This is because the hemisphere is half of the sphere.

Q: Can I use the formula for the volume of a hemisphere to find the volume of a sphere?

A: No, you cannot use the formula for the volume of a hemisphere to find the volume of a sphere. The formula for the volume of a hemisphere is V = (2/3)πr^3, which is different from the formula for the volume of a sphere, V = (4/3)πr^3.

Conclusion

In conclusion, the volume of a hemisphere inside a rectangular prism and a sphere can be calculated using the formulas V = (2/3)πr^3 and V = (4/3)πr^3, respectively. We have answered some frequently asked questions (FAQs) related to the volume of a hemisphere inside a rectangular prism and a sphere.

References

[1] "Mathematics for Dummies" by Mary Jane Sterling
[2] "Calculus for Dummies" by Mark Ryan
[3] "Geometry for Dummies" by Mark Ryan

Mathematical Formulas

Volume of a hemisphere: V = (2/3)πr^3
Volume of a sphere: V = (4/3)πr^3

Mathematical Symbols

π (pi): a mathematical constant representing the ratio of a circle's circumference to its diameter
r: the radius of a hemisphere or a sphere
V: the volume of a hemisphere or a sphere