What is Volume of a Hemisphere?

S3-SA2-0140

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The volume of a hemisphere is the amount of space it occupies. A hemisphere is exactly half of a sphere, like cutting a cricket ball into two equal halves. So, its volume is half the volume of a full sphere.

Simple Example

Quick Example

Imagine you have a round 'katori' (bowl) that is shaped like a hemisphere. If you fill this katori completely with water, the amount of water it holds is its volume. If a full spherical 'ladoo' has a volume of 100 cubic centimeters, then a perfect half-ladoo (hemisphere) would have a volume of 50 cubic centimeters.

Worked Example

Step-by-Step

PROBLEM: Find the volume of a hemisphere whose radius is 7 cm. (Use pi = 22/7)

STEP 1: Recall the formula for the volume of a sphere: V_sphere = (4/3) * pi * r^3.
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STEP 2: The formula for the volume of a hemisphere is half of a sphere's volume: V_hemisphere = (1/2) * V_sphere = (1/2) * (4/3) * pi * r^3, which simplifies to V_hemisphere = (2/3) * pi * r^3.
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STEP 3: Identify the given radius (r) = 7 cm. Substitute the value of r and pi into the formula.
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STEP 4: V_hemisphere = (2/3) * (22/7) * (7)^3
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STEP 5: V_hemisphere = (2/3) * (22/7) * (7 * 7 * 7)
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STEP 6: Cancel out one '7' from the numerator and denominator: V_hemisphere = (2/3) * 22 * (7 * 7)
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STEP 7: V_hemisphere = (2/3) * 22 * 49
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STEP 8: V_hemisphere = (44 * 49) / 3 = 2156 / 3 = 718.67 cubic cm (approximately).

ANSWER: The volume of the hemisphere is approximately 718.67 cubic centimeters.

Why It Matters

Understanding volume helps engineers design efficient water tanks or fuel containers. In AI/ML, it's used in 3D object recognition, while in Physics, it's crucial for calculating buoyancy or fluid displacement. Architects use it to plan space efficiently in buildings.

Common Mistakes

MISTAKE: Using the formula for a full sphere instead of a hemisphere. | CORRECTION: Always remember to multiply the sphere's volume formula by (1/2) or use the direct hemisphere formula (2/3) * pi * r^3.

MISTAKE: Confusing radius (r) with diameter (d). | CORRECTION: The formula uses radius. If diameter is given, divide it by 2 to get the radius before substituting into the formula (r = d/2).

MISTAKE: Forgetting to cube the radius (r^3). | CORRECTION: The volume formula involves r multiplied by itself three times. Double-check your calculation to ensure you've cubed the radius correctly.

Practice Questions

Try It Yourself

QUESTION: A hemispherical 'lota' (small pot) has a radius of 3 cm. What is its volume? (Use pi = 3.14) | ANSWER: Approximately 56.52 cubic cm.

QUESTION: If the diameter of a hemispherical dome is 14 meters, what is the volume of air inside it? (Use pi = 22/7) | ANSWER: Approximately 718.67 cubic meters.

QUESTION: A solid toy is in the form of a hemisphere mounted on a cone. If the radius of the hemisphere is 4 cm and the total volume of the toy is 200 cubic cm, what is the volume of the conical part? (Use pi = 3.14) | ANSWER: Approximately 107.33 cubic cm.

MCQ

Quick Quiz

What is the formula for the volume of a hemisphere with radius 'r'?

(4/3) * pi * r^3

(2/3) * pi * r^3

pi * r^2

2 * pi * r

The Correct Answer Is:

The volume of a hemisphere is half the volume of a sphere. The volume of a sphere is (4/3) * pi * r^3, so half of that is (1/2) * (4/3) * pi * r^3, which simplifies to (2/3) * pi * r^3.

Real World Connection

In the Real World

In India, many water tanks on rooftops are hemispherical or have hemispherical bottoms to store water efficiently. Engineers calculate the volume of these tanks to know how much water they can hold, which is important for urban planning and water supply management in cities.

Key Vocabulary

Key Terms

HEMISPHERE: Half of a sphere | VOLUME: The amount of space an object occupies | RADIUS: Distance from the center to the edge of a circle or sphere | PI (pi): A mathematical constant, approximately 3.14 or 22/7 | CUBIC UNITS: Units used to measure volume, like cubic centimeters or cubic meters

What's Next

What to Learn Next

Great job learning about hemisphere volume! Next, you can explore the 'Surface Area of a Hemisphere' to understand how much material is needed to cover its outer part. This builds on your understanding of its dimensions.