Volume of Hemisphere with Formula and Worked Examples
Volume of hemisphere is the space occupied by half of a sphere. Its volume is calculated using V = (2/3)πr³, where r is the radius. This page covers the formula, diameter-based calculations, unit conversions, solved examples and common aptitude problems involving hemispherical solids.
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What Is the Volume of a Hemisphere?
A hemisphere is formed by cutting a sphere through its centre. Since a sphere has volume (4/3)πr³, one hemisphere has half of this volume: (1/2) × (4/3)πr³ = (2/3)πr³. The answer is expressed in cubic units, such as cm³ or m³.
Volume of Hemisphere Video Lessons
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Sphere and Hemisphere: Volume and Surface Area
Learn the formulas and methods to calculate the volume and surface area of spheres and hemispheres, including how to apply them to quantitative aptitude problems.
Volume of Hemisphere Formula & Tricks
Important Formulas
Use this formula when the radius r of the hemisphere is known.
Since r = d/2, substituting it into (2/3)πr³ gives V = πd³/12.
Use this rearranged formula when the volume is given and the radius is required.
This formula gives the diameter directly from the volume.
Quick Tricks
The main formula uses radius, not diameter. Divide the diameter by 2 before substituting.
When r is a multiple of 7, using π = 22/7 often allows direct cancellation and faster calculation.
A hemisphere has exactly half the volume of a sphere with the same radius. Thus, two identical hemispheres have the volume of one sphere.
Volume of Hemisphere Concepts
Applying the Hemisphere Volume Formula
The radius must be in the same unit as the required answer. If r = 3 cm, then V = (2/3) × π × 3³ = 18π cm³, which is approximately 56.55 cm³ using π ≈ 3.1416.
Volume When Diameter Is Given
The diameter is twice the radius. For d = 12 cm, r = 6 cm, so V = (2/3)π(6³) = 144π cm³. The direct formula gives the same result: π(12³)/12 = 144π cm³.
Relation Between Hemisphere and Sphere
Sphere volume = (4/3)πr³, while hemisphere volume = (2/3)πr³. Therefore, sphere volume : hemisphere volume = 2 : 1. This relation applies only when both solids have the same radius.
Finding Radius from the Given Volume
The volume must be substituted in cubic units. If V = 36π cm³, then r³ = 3(36π)/(2π) = 54, so r = ∛54 cm, approximately 3.78 cm.
Capacity and Cubic Unit Conversion
For larger units, 1 m³ = 1,000 L. Convert the calculated volume only after using consistent length units in the formula.
Volume of Hemisphere Visual Guide & Data
Use the table and chart below to understand the data better.
Common Volume Unit Conversions
Equivalent units used for hemispherical volume and capacity calculations.
| Volume unit | Equivalent value |
|---|---|
| 1 cm³ | 1 mL |
| 1,000 cm³ | 1 L |
| 1 m³ | 1,000 L |
| 1 L | 1,000 mL |
Note: Use cubic units for geometric volume and litre or millilitre units for liquid capacity.
Radius of a Hemisphere
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Volume of Hemisphere Revision Points
Use these formulas and rules for quick revision.
- Volume of a hemisphere = (2/3)πr³.
- If diameter d is given, volume = πd³/12.
- Radius = diameter/2.
- A hemisphere has half the volume of a sphere with the same radius.
- Volume is measured in cubic units.
- 1 cm³ = 1 mL, 1,000 cm³ = 1 L, and 1 m³ = 1,000 L.
- Use the same length unit for radius and the required volume unit.
Volume of Hemisphere FAQs
What is the volume of a hemisphere with radius 7 cm?
V = (2/3) × (22/7) × 7³ = 2156/3 cm³, or approximately 718.67 cm³.
What is the volume formula for a hemisphere when its diameter is given?
The formula is V = πd³/12, where d is the diameter.
What is the volume of a hemisphere with diameter 10 cm?
The radius is 5 cm. Therefore, V = (2/3)π(5³) = 250π/3 cm³, approximately 261.80 cm³.
How is the volume of a hemisphere related to the volume of a sphere?
For the same radius, the volume of a hemisphere is half the volume of the sphere: hemisphere volume = sphere volume/2.
What is the radius of a hemisphere whose volume is 36π cm³?
Using r = ∛(3V/(2π)), r = ∛54 cm, which is approximately 3.78 cm.
What unit is used for the volume of a hemisphere?
Volume is expressed in cubic units such as cm³, m³ or mm³. For liquid capacity, 1 cm³ equals 1 mL.