Volume of Hemisphere
In geometry, we have studied different types of three-dimensional shapes. In three-dimensional shapes, the solids have three different measurements or dimensions like length, breadth and height. We know that 3D shapes do not lie on a piece of paper. Usually, most of the three-dimensional objects are obtained from the rotation of the two-dimensional objects.
One of the best examples of the 3D shape is the sphere which is obtained from the rotation of 2D shape called the circle. Our earth is a good example of a sphere which is spherical.
Hemisphere Definition
A sphere is defined as a set of points in three-dimension, and all the points lying on the surface is equidistant from the centre. When a plane cuts across the sphere at the centre or equal parts, it forms a hemisphere. We can say, a hemisphere is exactly half of a sphere. In general, a sphere makes exactly two hemispheres. One such good example of the hemisphere is our earth. Our earth consists of two hemispheres, namely Southern Hemisphere and the Northern Hemisphere.
Hemisphere Volume
We can easily find the volume of the hemisphere since the base of the sphere is circular. The volume of the hemisphere is derived by Archimedes.
The volume of a hemisphere = (2/3)πr3 cubic units.
Where π is a constant whose value is equal to 3.14 approximately.
“r” is the radius of the hemisphere.
Hemisphere Equation
When the radius “R” is centred at the origin, it is given by
x2 + y2 + z2 = R2
The Cartesian form or equation of a hemisphere with the radius “R” 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 follows
x = r cos θ sin ∅
y = r sin θ cos ∅
z = r cos ∅
Solved Problem
A sample problem on hemisphere is given below
Question:
Find the volume of the hemisphere whose radius is 6 cm.
Solution:
Given:
Radius, r = 6 cm
The volume of a hemisphere = (2/3)πr3 cubic units.
Substitute the value of r in the formula.
V = (2/3) × 3.14 × 6 × 6 × 6
V = 2× 3.14 × 2 × 6 × 6
V = 452.16
Therefore, the volume of the hemisphere is 452.16 cubic units.
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