Volume Of Sphere And Hemisphere

Volume of a Hemisphere refers to the capacity of a hemisphere. A hemisphere is the precise one-half of a sphere geometrically. As a outcome, a hemisphere's volume is one-half that of a sphere. The volume of a hemisphere is given by the formula (2/3)πr³ cubic units, where 'r' refers to the radius of the hemisphere and π is a constant whose value is equal to 22/7 or 3.14 approximately. The northern and southern halves of the Earth, the left and right halves of our brain, a basin, and an igloo are all examples of hemispheres in real life. The book of a hemisphere, similar its expanse, is an important mathematical calculation that aids in the solution of many mensuration problems.

Cardinal Terms: Hemisphere, Sphere, 3-Dimensional Shapes, Coordinates, Book, Area, Three-Dimensional Geometry, Radius

Definition of Hemisphere

[Click Here for Sample Questions]

A hemisphere is formed when a aeroplane cuts across the sphere at the centre or in equal portions. "Hemi" means half, and a sphere is defined equally a gear up of three-dimensional points, with all points on the surface being equidistant from the center.

In geometry, a hemisphere is a three-dimensional shape formed by splitting a sphere into 2 equal half, each with one apartment side. In real life, we come up across a variety of hemisphere-shaped objects. For instance, cutting around a watermelon in one-half yields a hemisphere-shaped watermelon, and cutting a cerise in half yields a hemisphere.

Hemisphere

Hemisphere

Read More than: Introduction to Three-Dimensional Geometry

Examples of Hemisphere

[Click Here for Sample Questions]

An igloo is a well-known example of a hemispheric object in everyday life. The curved confront of the hemisphere is formed past the acme of the igloo, while the flat face up is formed by the base or the ground.
The majority of the bowls are hemispheric in shape.
A curved surface and a flat base can be seen on the mushroom cap if you await attentively. Every bit a result, a mushroom is a archetype case of a daily hemisphere-shaped item.
A hat's shape is like to that of a hemisphere since information technology has a curved acme and a apartment base of operations.
The earcups of a headphone are an excellent illustration of hemispheric items seen in oursurround.

Volume of Hemisphere

[Click Here for Sample Questions]

The total capacity of a hemisphere is the number of unit cubes covered inside that area, and the volume of a hemisphere is the number of unit of measurement cubes covered within that infinite. The volume of a hemisphere is stated in cubic units, such equally thousand³, cm³, in³, and so on.

The book of a hemisphere equals the half thebook of a sphere, thus, it is expressed as,

Volume of Hemisphere = (two/three)πr ³

Where

'r' refers to the radius of the hemisphere
'π' is a constant whose value is equal to 22/7 or three.14 approximately.

Example: Calculate the volume of a hemisphere whose radius is half-dozen cm.

Solution: Radius (r) = half-dozen cm

Volume of Hemisphere = (2/iii)πrⁱⁱⁱ

= ii/iii × 3.14 × 6 × 6 × 6

= 452.16 cm³

Volume of Hemisphere

Volume of Hemisphere

Read More: Surface Area of A Hemisphere

Equation of Hemisphere

[Click Here for Sample Questions]

Let the distance from the centre i.e. the radius equally R

ten² + y² + z^two = Rⁱⁱ

The equation of a hemisphere with the radius "R" at the point (x₀, y₀, z₀) is written every bit

(x-ten₀)² + (y-y₀)ⁱⁱ + (z-z₀)² = R²

Then the spherical coordinates of the hemisphere will exist

x = r cos θ sin ∅
y = r sin θ cos ∅
z = r cos ∅

Read More than: Coordinate Geometry

Things to Remember

A hemisphere is a three-dimensional shape formed past splitting a sphere into two equal halves.
The volume of a hemisphere is half the volume of a sphere or we can say two hemispheres are combined to form a sphere.
The volume of the hemisphere is given every bit (two/iii)πrⁱⁱⁱ cubic units, where 'r' denotes the radius of the hemisphere and π is equal to 22/7 or 3.xiv approximately.
The spherical coordinates of the hemisphere are x = r cos θ sin ∅, y = r sin θ cos ∅ and z = r cos ∅.

Sample Questions

Ques. Find the volume of a hemisphere having a radius of 5 cm? (3 Marks)

Ans. The radius of the hemisphere = 5 cm

Volume of a hemisphere = Five = (two/3)πr³

= 2 × 3.14 × five × 5 × v / 3

= 261.67 cubic cm

Hence the volume of given hemisphere is 261.67 cubic cm.

Ques. The circumference of a hemispherical bowl's edge is 132 cm. Determine the capacity of the bowl. (three Marks)

Ans. Circumference of the bowl = 132 cm

Circumference = 2πr

2πr = 132 cm

r = 132/2π

r = 21 cm

Now, Book of Hemispherical basin = V = (ii/3)πr³

V = 2/3 × 22/7 × (21)³

= 19404 cm^three

Hence the capacity of the bowl is 19404 cm³

Ques. What is the relation betwixt the book of a sphere and the volume of a hemisphere? (three Marks)

Ans. A 3-dimensional solid's volume is the amount of infinite it takes upwards. Cubic units are used to measure volume i.e. cm^three, g^three, etc. Earlier computing the volume, it should be fabricated certain that all of the measurements are in the same unit.

Volume of Sphere = four/3πrⁱⁱⁱ

A hemisphere'southward volume is one-half of the book of the associated sphere.

Book of Hemisphere = 2/3πr^three

Ques. Define Hemisphere in three-dimensional geometry. (2 Marks)

Ans. In general, a hemisphere refers to half of the earth, such every bit the northern or southern hemisphere. In geometry, yet, a hemisphere is a three-dimensional shape formed by splitting a sphere into two equal half, each with i flat side.

Ques. What are the curved surface surface area and total surface expanse of a hemisphere? (three Marks)

Ans. The curved surface area refers to the area of the hemisphere'south outer surface. The overall surface area includes the area of the curved surface every bit well equally the surface area of the circular (base).

The curved area of a hemisphere = 2πr² square units
The total surface area of a hemisphere = 3πr² square units

Ques. A hemisphere has a volume of 2500 cm ³ . Calculate the radius of the hemisphere? (3 Marks)

Ans. The volume of a given hemisphere = 2500 cubic cm

Book of a hemisphere = V = (two/3)πrⁱⁱⁱ

2500 = (ii/3)πr³

2500 × 3 / 2π = r³

r³ = 7500/2π

rⁱⁱⁱ = 1194.267

r = ⁱⁱⁱ √(1194.267)

r = 10.61 cm

Hence, the radius of given hemisphere is x.61 cm.

Ques. A sphere with a radius of 4 cm is divided into two equal one-half. Calculate the book of each produced hemisphere. (three Marks)

Ans. Radius of Sphere = 4cm

And then, radius of Hemisphere = 4cm

Book of each Hemisphere = V = (2/iii)πr³

5 = (ii/3)πr³

5 = (2/three)π 4³

Five = 133.nine cubic cm

Hence the book of each Hemisphere is 133.9 cubic centimetres.

Ques. For a hemisphere whose centre is at the origin, what is the cartesian equation? (3 Marks)

Ans. x^two + y^two + z² = r^two is the cartesian equation for a hemispheric shape with its centre at the origin.

And, the spherical coordinates of the hemisphere will be

x = r cos θ sin ∅
y = r sin θ cos ∅
z = r cos ∅

Ques. What is the volume of a sphere having a radius of 3 cm? (3 Marks)

Ans. Radius of sphere r = 3cm

Volume of a sphere = V = 4πr³/3 cubic units

V = 4/3 × 3.14 × 3³

5 = 4/three × 3.14 × 3 × 3 × 3

V = 113.04 cm^three

Hence the volume of given sphere is 113.04 cubic cm.

Ques. What is the volume of a sphere having a diameter of 10 cm? (three Marks)

Ans. The diameter of given sphere = 10 cm

Then the radius = diameter/2 = ten/ii = 5 cm

Now, for volume of sphere,

Book = Five= 4πr³/3 cubic units

Five = four/three π 5ⁱⁱⁱ

V = four/iii × 22/7 × v × 5 × v

V = 4/3 × 22/7 × 125

V = 523.viii cmⁱⁱⁱ

Hence the book of given sphere is 523.8 cubic cm.

Ques. A airplane passes through the centre of a sphere, forming a 12-foot-radius circle. What is the sphere's surface area? (ii Marks)

Ans. The radius of the given sphere r = 12 anxiety

Total surface area of the sphere (V) = four π r²

= 4 × 3.14 × 12 × 12

= 1808.64

Hence surface area of the sphere is 1808.64 square feet.

Ques. A spherical object has a surface expanse of 379.94 thou ² . What is the object'southward bore? (3 Marks)

Ans. Surface area of given sphere = 379.94 mⁱⁱ

And surface area of a sphere = 4πr²

Now,

379.94 = four × 3.xiv × r²

379.94 = 12.56 r²

rⁱⁱ = 379.94/12.56

rⁱⁱ = 30.25

r = √xxx.25

r = v.5

Therefore, the radius of the spherical solid is five.5 k.

As well Cheque: