Volume of Hemisphere

While studying three-dimensional geometry, the concept of the hemisphere is an important one. Hemispheres are a three-dimensional shape that forms half of a sphere. It has a flat base and a curved surface that extends upwards to resemble a sphere. Calculating the hemisphere volume is an essential mathematical application in physics, engineering, and everyday life. Here, we will explore the hemisphere volume formula, its derivations, practical examples, and how to use a hemisphere volume calculator to simplify the calculations. 

1.0What is a Hemisphere?

A hemisphere is a three-dimensional geometric shape that is made up of half a sphere, with one side being flat and the other having a bowl-size shape. When you cut a sphere into two halves along a plane that passes through the centre, each half is called a hemisphere. Understanding the hemisphere volume formula can help us understand the capacity it can hold, which is especially useful in real-world scenarios like designing domes, bowls, and liquid storage tanks. 

2.0Hemisphere Volume Formula

We can easily find the volume of a hemisphere since the base of the hemisphere is circular. The hemisphere volume formula is derived by Archimedes. If the radius of the hemisphere is “r,” then the formula for the volume of the hemisphere would be: 

V = (2/3)πr³ cubic units

Here, “V” is the volume of the hemisphere, “r” is the radius of the hemisphere, and π is a mathematical constant (approximately 3.14159). 

This hemisphere volume formula allows us to determine the space occupied by the hemisphere with the help of the radius.

3.0Derivation of Volume of Hemisphere

To understand the derivation of the volume of the hemisphere, one needs to have knowledge of calculus and integration. The volume of a solid can be determined using the disk method. In theory, it involves cutting up the hemisphere into thin, circular disks and summing up their volume. Inherently, it confirms the hemisphere volume formula. 

4.0Volume of Hemisphere Example

Let’s apply the hemisphere volume formula to the real-life volume of hemisphere examples to understand the concept in a more in-depth manner. 

Question: Find the volume of the hemisphere whose radius is 6 cm.

Solution: Radius, r = 6 cm

The volume of any hemisphere is (2/3)πr³ cubic units

Substitute the value of r = 6 in the formula.

V = (2/3) × 3.14 × 6 × 6 × 6

V = 2× 3.14 × 2 × 6 × 6

V = 452.16 cm³

Question: Find the volume of a hemisphere of diameter 5 cm.

Solution: Diameter = 5 cm

Thus, r = 5/2 [Diameter = 2 (Radius)]

Volume of Hemisphere = 2/3 π r³

⇒ Volume = 2/3 π (5/2)³

⇒ Volume = 32.724 cm³

Question: If the volume of the hemisphere is 2.095 m³. Find the radius of the hemisphere.

Solution: Volume of hemisphere = 2/3 π r3

2.095 = 2/3 π r³

2.095 = 2.095 r³

r³ = 1

r = 1 m 

So, the radius of the hemisphere is 1 m.

5.0Practice Problems on Volume of Hemisphere

1. Find the volume of a hemisphere with a radius of 7 cm.
2. A hemispherical tank has a radius of 10 meters. How much water can it hold in cubic meters?
3. A hemispherical bowl has a diameter of 12 inches. Calculate its volume.
4. Compare the volumes of two hemispheres, one with a radius of 3 cm and the other with a radius of 6 cm. Which one has a greater volume?
5. A hemispherical dome is being constructed with a radius of 20 feet. How much volume does the dome occupy?

6.0Using a Hemisphere Volume Calculator

While manual calculators are useful for understanding the concept, a hemisphere volume calculator makes the process faster in cases of lengthy sums and equations. The process is swift and error-free, which leads to better outcomes. Hemisphere volume calculator is widely available online to help with complex mathematical problems and real-world applications. 

7.0Conclusion

The hemisphere volume formula is a fundamental concept in three-dimensional geometry. By understanding the concepts and applying them to real-world situations, one can easily use them in mathematical problems and real-world applications. Go through the examples and apply your understanding of the practice problems to excel in them.