Volume of Hemisphere Formula Derivation and Solved Examples
The concept of volume of hemisphere plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are calculating the space inside a bowl, dome, or a water tank shaped like a half-sphere, understanding how to find the volume of a hemisphere is essential in both academics and day-to-day problem-solving.
What Is Volume of Hemisphere?
A hemisphere is a three-dimensional solid that represents exactly half of a sphere. When you cut a perfect sphere into two equal parts through its center, each half is called a hemisphere. Real-world examples include the two hemispheres of the Earth, domes, bowls, and some types of tanks. This idea connects to concepts like sphere volume, cylinder volume, and cone volume in geometry.
Key Formula for Volume of Hemisphere
Here’s the standard formula: \( V = \frac{2}{3} \pi r^3 \)
Where:
- \( r \) = radius of the hemisphere
- \( \pi \) (pi) ≈ 3.14 or \( \frac{22}{7} \)
Cross-Disciplinary Usage
The volume of hemisphere formula is not only useful in Maths but also plays an important role in Physics (calculating displacement by a floating hemisphere), Computer Science (graphics and modeling), and daily logical reasoning (estimating capacities of tanks or domes). Students preparing for JEE, NEET, or competitive exams will often find questions based on this concept.
Step-by-Step Illustration
Let’s solve a problem using the volume of hemisphere formula with both radius and diameter.
Find the volume of a hemisphere with a radius of 5 cm. Take \( \pi = 3.14 \).
1. Identify the given radius: r = 5 cm
2. Apply the formula:
3. Substitute values:
4. Calculate \( 5^3 = 125 \):
5. Plug into equation:
6. \( 3.14 \times 125 = 392.5 \):
7. \( \frac{2}{3} \times 392.5 = 261.67 \):
8. Final Answer: The volume is 261.67 cm³.
Volume of a hemisphere with diameter 12 cm.
1. Diameter = 12 cm ⇒ radius r = 12/2 = 6 cm
2. Apply the formula:
3. Calculate \( 6^3 = 216 \):
4. \( V = \frac{2}{3} \times 3.14 \times 216 \)
5. \( 3.14 \times 216 = 678.24 \):
6. \( \frac{2}{3} \times 678.24 = 452.16 \):
7. Final Answer: Volume = 452.16 cm³.
Speed Trick or Vedic Shortcut
If you’re given the diameter (d), simply halve it to get the radius (r = d/2), then go directly to the formula for the volume of hemisphere without recalculating. Always remember to match your units (cm, m, etc.) before starting calculations to avoid silly mistakes. Vedantu’s live classes often share such tips to boost students’ calculation speed.
Try These Yourself
- Find the volume of a hemisphere with radius 8 cm. (π = 3.14)
- A bowl in the shape of a hemisphere has diameter 10 cm. What is its volume?
- If the volume of a hemisphere is 904.32 cm³, what is its radius? (π = 3.14)
- Calculate the volume, in litres, of a hemispherical tank with radius 0.5 m.
Frequent Errors and Misunderstandings
- Mixing up the formulas for volume and surface area of a hemisphere.
- Confusing radius with diameter—always halve the diameter to get radius.
- Forgetting to cube the radius (using r² instead of r³).
- Not expressing answers in cubic units (like cm³ or m³).
- Plugging in the wrong value of π; use 3.14 or 22/7 unless told otherwise.
Relation to Other Concepts
The idea of volume of hemisphere connects closely with sphere volume, cylinder volume, and cone volume. Understanding how to move from 2D (circle area) to 3D (solid volumes) will help you master topics in geometry, physics, and engineering. Practice also with the maths surface area and volume collection for revision.
Classroom Tip
A simple way to remember the volume of hemisphere formula is: “Half-sphere means two-thirds pi r-cube” or “V = 2/3 π r³.” Drawing a quick sketch of a half-ball with a flat circular base makes the concept stick in memory. Vedantu educators use similar visuals in live classes to make formulas easy and exam-friendly.
Wrapping It All Up
We explored the volume of hemisphere—from definition, formula, solved examples, common mistakes, and connections to related chapter concepts. Regular practice with Vedantu’s resources helps you gain confidence for both school tests and real-world applications.
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FAQs on Volume of a Hemisphere Explained with Formula and Applications
1. What is the volume of a hemisphere?
The volume of a hemisphere is the amount of space inside half of a sphere and is given by the formula V = (2/3)πr³, where r is the radius. A hemisphere is exactly half of a sphere, so its volume is half of (4/3)πr³. This formula is widely used in mensuration and solid geometry problems.
2. What is the formula for the volume of a hemisphere?
The formula for the volume of a hemisphere is V = (2/3)πr³. Here:
- r = radius of the hemisphere
- π ≈ 3.14 or 22/7
3. How do you calculate the volume of a hemisphere step by step?
To calculate the volume of a hemisphere, use the formula V = (2/3)πr³ and substitute the radius value.
- Step 1: Find the radius (r).
- Step 2: Calculate r³.
- Step 3: Multiply by π.
- Step 4: Multiply by 2/3.
4. What is the volume of a hemisphere with radius 7 cm?
The volume of a hemisphere with radius 7 cm is (2/3)π(7³) = (686/3)π cm³.
- r = 7 cm
- r³ = 343
- V = (2/3)π × 343
5. Why is the volume of a hemisphere half the volume of a sphere?
The volume of a hemisphere is half the volume of a sphere because a hemisphere is formed by cutting a sphere into two equal halves. Since the sphere’s volume is (4/3)πr³, dividing it by 2 gives (2/3)πr³. This geometric relationship explains the formula directly.
6. What is the difference between the volume and surface area of a hemisphere?
The volume of a hemisphere measures the space inside it, while the surface area measures the area covering its surface.
- Volume formula: (2/3)πr³
- Curved surface area: 2πr²
- Total surface area: 3πr²
7. How do you find the radius if the volume of a hemisphere is given?
To find the radius from the volume of a hemisphere, rearrange the formula V = (2/3)πr³ to get r = ∛(3V / 2π).
- Step 1: Multiply V by 3.
- Step 2: Divide by 2π.
- Step 3: Take the cube root.
8. What units are used for the volume of a hemisphere?
The volume of a hemisphere is measured in cubic units such as cm³, m³, or in³. Since volume represents three-dimensional space, the unit must be cubed. For example, if radius is in centimeters, the volume will be in cubic centimeters (cm³).
9. Can you give a real-life example of the volume of a hemisphere?
A real-life example of the volume of a hemisphere is calculating the capacity of a hemispherical bowl or dome. If a bowl has radius 10 cm, its volume is (2/3)π(10³) = 2094.67 cm³ (approx). This helps determine how much liquid the bowl can hold.
10. What are common mistakes when calculating the volume of a hemisphere?
Common mistakes when calculating the volume of a hemisphere include using the wrong formula or forgetting to cube the radius.
- Using (4/3)πr³ instead of (2/3)πr³
- Not calculating r³ correctly
- Confusing radius with diameter
- Using incorrect units
