MathsVolume of Hemisphere

Volume of Hemisphere

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.

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    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 Hemisphere

    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.

    1. Curved Surface Area
    2. 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.

    1. Curved Surface Area
    2. 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.

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