Title:How to Find the Volume of a Hemisphere

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Introduction:-

Realizing the concept of volume is crucial in mathematics

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Q: How to find the volume of a hemisphere with diameter

Answer: ∵ Radius of hemisphere(r)= d/2 where, d is the diameter of hemisphere Now, Volume of hemisphere = ⅔ πr³ = ⅔ π(d/2)³ = ⅔ πd³/8 = 1/12 πd³

Q: What is the volume and SA of a hemisphere?

Answer: Volume of hemisphere = ⅔ πr³ where, r = Radius of the hemisphere; and SA of a hemisphere = CSA of hemisphere + Area of its base = 2πr² + πr² = 3πr², where, r = Radius of the hemisphere

Q: What is a hemisphere in maths?

Answer: In maths, a hemisphere is defined as half of a sphere, which is divided by a plane passing through its center. It can be thought of as the upper or lower half of a sphere.

Q: What is the volume of a hollow hemisphere Class 10?

Answer: The volume of a hollow hemisphere = ⅔ π(R³- r³) where, R = Outer radius of sphere and r = Inner radius of hemisphere.

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and various practical applications. The volume of three-dimensional objects is particularly significant in fields like engineering, architecture, and physics. One of the shapes we often encounter is the hemisphere, which is half of a sphere.

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Today in this article I will try to provide a detailed explanation of How to Find the Volume of a Hemisphere, exploring its mathematical foundation, practical applications, and step-by-step calculations.

Explanation:-

To start today's article How to Find the Volume of a Hemisphere, Let’s first know,

What is a Hemisphere?

A hemisphere or a half sphere is defined as half of a sphere, which is divided by a plane passing through its center. It can be thought of as the upper or lower half of a sphere. The term “hemisphere” comes from the Greek word “hemisphairion,” meaning "half a sphere."

Properties of a Hemisphere:

Before diving into the calculations, it’s essential to understand the basic properties of a hemisphere:

Radius (r): The radius is the distance from the center of the hemisphere to any point on its curved surface.

Base: The flat circular surface at the base of the hemisphere is known as its base. The area of this circular base can be calculated using the formula:

Area of base= πr²

Height: The height of the hemisphere is equal to its radius.

Curved Surface Area: The curved surface area (CSA) of a hemisphere can be calculated using the formula:

CSA= 2 x Area of base

= 2 x πr²

= 2πr²

Total Surface Area: The total surface area (TSA) of a hemisphere, which includes the curved surface area and the area of the base, can be calculated as:

TSA = CSA + Area of base

= 2πr² + πr²

= 3πr²

Some Problems related on area of Hemisphere:

Question(1): The diameter of a hemisphere is 42cm, find its curved surface area.(Take π = 22/7)

FAQ

Answer:

Diameter of hemisphere (d) = 42 cm

∴ Radius of hemisphere(r) = d/2

= 42/2

= 21 cm

∴ CSA of the hemisphere = 2πr²

= 2 x 22/7x 21²

= 2 x 22/7 x 21 x 21

= 2 x 22 x 3 x 21

= 2772 cm²

Question(2):The radius of a hemisphere is 7m find its CSA.(Take π = 22/7)

FAQ

Answer:

Radius (r) = 7m

∴ CSA of the hemisphere = 2πr²

= 2 x 22/7 x 7²

= 2 x 22 x 7

= 308 m²

Question(3):The radius of a hemisphere is 10 cm,find its total surface area.(Take π = 3.14)

FAQ

Answer:

Radius of the hemisphere (r)=10cm

∴ TSA of hemisphere = 3πr²

= 3 x 3.14 x 10²

= 3 x 3.14 x 100

= 3 x 314.00

= 942cm²

Applications of Hemispheres:

Hemispheres are commonly found in various real-world fields, such as:

Architecture: Domes are often hemispherical in shape.

Sports: Many sports balls, like basketballs and soccer balls, can be watched as spheres, and their halves (hemispheres) are used in training tools.

Physics: In experiments involving projectiles and forces, hemispherical shapes can simplify be utilized.

Medical Science: The structure of certain organs, such as the human brain, can be approximated as hemispheres

Volume of a Hemisphere: Mathematical Foundation

The volume of a hemisphere can be derived from the volume of a sphere. The formula for the volume of a sphere is given by:

V = 4/3 πr³ where, r = Radius of the sphere

Since a hemisphere is half of a sphere, its volume can be calculated by simply dividing the volume of a sphere by 2:

Volume of Hemisphere=½ x 4/3 πr³

This simplifies to:

Volume of Hemisphere = ⅔ πr³

Key Variables in the Formula of Hemisphere:

V: Volume of the hemisphere.

r: Radius of the hemisphere.

π(pi): A mathematical constant approximately equal to 3.14159.

Step-by-Step Calculation of Volume of a Hemisphere:

Let’s walk through the process of calculating the volume of a hemisphere with a specific example.

Example: Calculate the Volume of a Hemisphere with a Radius of 5 cm

Step 1: We will first Identify the radius

In our example, the radius r of the hemisphere is given as 5 cm.

Step 2: We will substitute the radius into the volume formula

Using the volume formula:

V = ⅔ πr³

Substituting r = 5

V = ⅔ πr³

= ⅔π5³

Step 3: We will calculate r³

First, we will calculate 5³:

5³ =125

Step 4: We will substitute r3r^3r3 back into the formula

Now,we will substitute 125 into the formula:

V=2/3π(125)

Step 5: We will multiply ⅔ and 125

Calculating (2×125) / 3

250 /3 ≈ 83.333….≈ 83.33(approx.)

Step 6: Lastly we will include π(pi) in the final calculation

Finally, multiply by π(pi):

V≈83.33π ≈ 83.33 x 3.14 cm3(using π≈3.14) ∴V ≈ 261.6562

Thus, the volume of the hemisphere with a radius of 5 cm is approximately 261.65 cm³.

Some Problems related on Volume of Hemisphere:

(1) Question: Find the volume of the hemisphere whose radius is 21 cm.(Take π = 22/7)

FAQ

Answer:

Radius of the hemisphere (r)=21cm

∴ Volume of the hemisphere = ⅔ πr³

= ⅔ x 22/7 x (21)³

= ⅔ x 22/7 x 21 x 21 x 21

= 2 x 22 x 21 x 21

= 19404 cm³

(2) Question: Find the radius of the hemisphere whose volume is 18π m³.

FAQ

Answer:

Volume of the sphere = 18 π m³

Let, the radius of the hemisphere = r meter

A/Q,

⅔ πr³ = 18 π

=> r³ = (18π x 3) / 2π

=> r³ = 9 x 3

=> r³ = 3³

=> r = 3

∴ The radius of the hemisphere(r) = 3 m

(3) Question: The diameter of a hemisphere is 60cm, find it’s volume.(Take π = 3.14)

FAQ

Answer:

Diameter of hemisphere (d) = 60cm

∴ Radius of hemisphere(r) = d/2

= 60 /2

= 30 cm

∴ Volume of the hemisphere = ⅔ πr³

= ⅔ x 3.14 x 30³

= ⅔ x 3.14 x 27000

= 2 x 3.14 x 9000

= 56520 cm³

Visualizing a Hemisphere:

To enhance understanding, it can be helpful to visualize a hemisphere and its dimensions. Let,us Imagine a bowl or a dome-shaped object, where the curved surface represents the outer part of the hemisphere and the flat circular part represents the base.

Comparing the Volume of a Hemisphere to Other Shapes:

Volume of a Cylinder:-

To better understand the significance of a hemisphere’s volume, let’s compare it to a cylinder with the same radius and height. The formula for the volume of a cylinder is:

V = πr²h

For a cylinder we have taken a radius of 5 cm and a height of 5 cm:

V=π(5²)(5)=π(25)(5)=125π =125 x 3.14 [Taking π = 3.14] ≈392.50 cm³(approx.)

Volume of a Cone:-

Similarly, the volume of a cone is given by:

V = ⅓ πr²h

For a cone also same measurement i.e; radius of 5 cm and a height of 5 cm:

V=⅓ π(5²)(5)=⅓ π(25)(5)=125/3π = 41.66 x 3.14[taking π = 3.14] ≈130.81 cm³

Comparative Analysis:-

Hemisphere Volume: Approximately 261.65 cm³ (we have done in example)

Cylinder Volume: Approximately 392.50 cm³

Cone Volume: Approximately 130.81 cm³

From this comparison, we can observe that the volume of a hemisphere is greater than that of a cone but less than that of a cylinder with the same dimensions.

Common Mistakes When Calculating Volume of a Hemisphere:

While calculating the volume of a hemisphere may seem straightforward, there are common mistakes that learners often encounter:

Incorrect Formula Usage: Using the formula for the volume of a sphere instead of the hemisphere can lead to confusion. We should remember that the volume of a hemisphere is half that of a sphere.

Miscalculating the Radius: We must ensure that the radius is accurately measured or provided. Confusion can arise if the diameter is mistakenly used as the radius.

Neglecting Units: We always include units in your final answer. Failing to do so can lead to ambiguity in interpretation.

Rounding Errors: Be cautious when rounding off values, especially for π(pi). Depending on the context, rounding too early can lead to significant errors in the final result.

Real-World Applications of Volume Calculations:-

The calculation of the volume of a hemisphere has practical implications in various fields.

Here are some examples:

Engineering and Construction:-

In engineering, understanding the volume of hemispherical structures is crucial for:

Designing domes: Hemispherical shapes are common in architectural designs for roofs and other structures.

Reservoirs:Many water storage tanks are designed in the shape of a hemisphere to maximize volume while minimizing surface area.

Food and Beverage Industry:-

In the food industry, hemispherical molds are used to create desserts, such as:

Hemispherical cakes: Cakes molded in a hemisphere shape are popular for special occasions and celebrations.

Ice cream: Ice cream scoops often take on a hemispherical shape, and knowing the volume helps in packaging and serving sizes.

Environmental Science:-

In environmental studies, calculating the volume of hemispherical shapes can be relevant for:

Pollution studies: Understanding the dispersion of pollutants in water bodies can involve modeling hemispherical shapes.

Natural formations: Many geological features, such as certain types of hills or craters, may be approximated as hemispheres.

Conclusion:-

Understanding How to Find the Volume of a Hemisphere is a fundamental concept in geometry with wide-ranging applications. By mastering the formula V = ⅔ πr³,

we can accurately determine the volume of a hemisphere, facilitating calculations in various fields, from engineering to food science.

In this article, I have explored the definition and properties of a hemisphere, derived the volume formula, and provided step-by-step examples. I also discussed common mistakes to avoid and highlighted real-world applications, underscoring the importance of this geometric shape.

By applying this knowledge, we can enhance our mathematical skills and gain a deeper appreciation for the shapes and volumes we encounter in our daily lives. Whether we are an engineering student, a budding chef, or simply curious about geometry, the ability to calculate the Volume of a HemiSphere.

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FREQUENTLY ASKED QUESTIONs ON VOLUME OF HEMISPHERE:

FAQ

(a). What is hemisphere class 9?

Answer: A hemisphere is defined as half of a sphere, which is divided by a plane passing through its center. It can be thought of as the upper or lower half of a sphere. The term “hemisphere” comes from the Greek word “hemisphairion,” meaning "half a sphere."

(b). How to find the volume of a hemisphere with diameter

Answer:∵ Radius of hemisphere(r)= d/2

where, d is the diameter of hemisphere

Now,

Volume of hemisphere = ⅔ πr³

= ⅔ π(d/2)³

= ⅔ πd³/8

= (1/12) πd³

(c). What is the volume and SA of a hemisphere?

Answer:Volume of hemisphere = ⅔ πr³

where r = radius of the hemisphere

and

SA of a hemisphere = CSA of hemisphere + Area of its base

= 2πr² + πr²

= 3πr², where, r = Radius of the hemisphere.

(d). What is a hemisphere in maths?

Answer:In maths, a hemisphere is defined as half of a sphere, which is divided by a plane passing through its center. It can be thought of as the upper or lower half of a sphere.

(e).What is the volume of a hollow hemisphere Class 10?

Answer:The volume of a hollow hemisphere = ⅔ π(R³- r³)

where, R = Outer radius of sphere

and r = Inner radius of hemisphere.