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In geometry, we have a huge number of shapes. It might be two-dimensional or three-dimensional. We know that the 3D shapes are not drawn on the plane. Cube, cuboid, sphere, cone and more are great examples of 3D shapes. Shoe boxes, dusters, balls, and more are the real-life example of the 3D shapes around us.
What is a Hemisphere?
A hemisphere is formed when a plane intersects a sphere, dividing it into two equal parts. A sphere, defined as a set of equidistant points in three dimensions, generates precisely two hemispheres. Notably, the Earth exhibits this concept with its division into the Southern and Northern Hemisphere. Essentially, a hemisphere is half of a sphere, emphasising these three-dimensional structures’ symmetrical and divided nature.
Volume of Solid Hemisphere
A solid hemisphere is an object in the shape of exactly half a sphere. It is basically filled up with the material it is composed of. The volume of the solid hemisphere is easy to get. It uses exactly the same method used to derive the volume of the hemisphere.
Volume of Solid Hemisphere = (⅔)r3 cubic units
Volume of Hollow Hemisphere
A hollow hemisphere is an object in the shape of exactly half a sphere, but it has two diameters for the circular base. The first diameter is for the inner side of the circular base, and the second is for the outer circle.
Volume of hollow hemisphere formula = ⅔ 𝜋r3 – ⅔ 𝜋r3
Volume of Hemisphere Formula
The volume of the hemisphere can be derived directly by using a simple formula. The Volume of the Hemisphere formula is given below.
Volume of Hemisphere = (⅔)r3 cubic units
Volume of Hemisphere Formula Equation
When the radius of the hemisphere, r, is centred at the origin, it is given by
x2 + y2 + z2 = r2
The Cartesian of this hemisphere at the point (x0, y0, z0) is written as
(x-x0)2 + (y- y0)2 + (z- z0)2 = r2
Therefore, the spherical coordinates of the hemisphere are given as:
x = r cos θ sin ∅
y = r sin θ cos ∅
z = r cos ∅
How to Calculate the Volume of the Hemisphere?
The Volume of the Hemisphere is calculated using the formula,
Volume of Hemisphere formula = (⅔)r3 cubic units
Therefore, now let’s find the Volume of a hemisphere whose radius is 3cm.
Now, let’s follow the steps given below:
Step 1: Note the radius of the given hemispheres. In this case, the radius assumed is 3 cm. Therefore, the Volume of the Hemisphere = (⅔)r3 cubic units.
Step 2 : Substitute the value of radius in the formula,
Volume of the Hemisphere = (⅔) 33 cubic cm
Step 3 : Now, solve the values. Hence we get,
Volume of the Hemisphere = 18 cubic cm
Volume of the Hemisphere: Solved example
Ques 1. Find the volume of the hemisphere if the radius is 3 cm.
Ans. given radius = 3cm
Volume of hemisphere = (⅔)r³
Therefore volume of given hemisphere is,
Volume of hemisphere = (⅔) * 33 = 2 * 9 =18 cm cube
Ques 2. Find the volume of the hemisphere if the radius is 30 cm.
Ans. Given radius = 30 cm
Volume of hemisphere = (⅔)r³
Therefore volume of given hemisphere is equal to (⅔) * 303 = 2 * 9000 =18000 cm cube
More about Hemisphere
Curved Surface Area
A hemisphere possesses only one curved surface area.
Volume Formula
The volume of a hemisphere is given by the formula V = (2/3)πr³, where “r” is the radius, and π is approximately 3.14.
Surface Areas
A hemisphere has two surface areas: curved surface area (CSA) and total surface area (TSA).
The curved surface area represents the region covered by the curved surface.
The total surface area is the sum of the curved surface area and the base area.
Measurement Unit
Since volume involves three dimensions (length, width, and height), the volume of a hemisphere is measured in cubic units.
Spherical Coordinates
The spherical coordinates of a hemisphere are represented as (ρ, φ, θ), where ρ is the radial distance, φ is the polar angle, and θ is the azimuthal angle.
Hemisphere Equation
When the hemisphere’s radius “R” is centred at the origin, the equation is given by x² + y² + z² = R².
Cartesian Form
The Cartesian form or equation of a hemisphere with the radius “R” at the point (x₀, y₀, z₀) is x² + y² + z² = R².
Spherical Coordinates (Alternative)
The spherical coordinates of the hemisphere are expressed as (R, π/2, 0) in terms of radial distance, polar angle, and azimuthal angle.
Surface Area of Hemisphere
The Surface Area of Hemisphere can be easily calculated. Now, we have two kinds of surface area.
- Curved Surface Area
- Total Surface Area
Curved Surface Area of Hemisphere Formula
CSA of hemisphere= ½ Surface Area of the Sphere
CSA = ½ 4 πr²
CSA = 2πr²
Therefore, the Curved Surface Area of a Hemisphere is 2πr²
Total Surface Area of Hemisphere Formula
Total Surface Area of Hemisphere Formula is the sum of CSA and base area.
Therefore, TSA = 2πr² + πr²
TSA = 3πr²
Therefore, the Total Surface Area of the Hemisphere is equal to 3πr²
The Surface Area Of A Hemisphere: Solved Example
Ques. Find the surface area of a hemisphere whose radius is 7 cm?
Ans. Given: Radius = 7 cm
as we know, we have two kinds of surface area.
- Curved Surface Area
- Total Surface Area
Therefore, The curved surface area = 2πr2 unit square
and
The total surface area = 3πr2 unit square
So,
(i) CSA of the hemisphere= 2 × π × 7 × 7
CSA = 3.14 × 2 × 49
CSA = 308 cm2
(ii) TSA of the hemisphere = 3 × π × 7 × 7
TSA = 3.14 × 3 × 49
TSA = 462 cm2
Therefore, the curved surface area and the total surface area of the hemisphere are 308 and 462 cm2 , respectively.
FAQs on Volume of Hemisphere
What is a hemisphere, and how is it related to a sphere?
A hemisphere is formed when a plane intersects a sphere, dividing it into two equal parts. It represents half of a sphere, and examples of hemispheres from our surroundings include the Northern and Southern Hemispheres of the Earth
How is the volume of a hemisphere calculated, and what is the formula?
The volume of a hemisphere is calculated using the formula: Volume = (⅔)r³ cubic units, where 'r' is the radius of the hemisphere.
How can I calculate the volume of a hemisphere if the radius is known?
To calculate the volume of a hemisphere, use the formula: Volume = (⅔)r³ cubic units. For example, if the radius is 3 cm, substitute it into the formula to find the volume.
