Modeling the volume of a sphere :
A sphere is a three-dimensional figure where all the points on the sphere would be at the same distance from the center. The radius of a sphere is the distance between the center to any point on the sphere.
We already know that a cone can fill one-third of a cylinder of where the radius and height of the cone and cylinder would be same.
If we do a similar experiment with a sphere of the same radius that the cylinder has, we will find that the sphere can fill two-third of the cylinder.
Here, the cylinder’s height would be equal to two times the the radius of the sphere.
Step 1 :
Write the formula for volume V of a cylinder with base area B and height h.
V = B · h
Step 2 :
Find the base area B of the cylinder.
We know that the base of the cylinder is a circle (Look at the figure given below).
So, the area of the base of a cylinder is
B = ∏r²
Step 3 :
Write the formula for volume V of a cylinder with base area B = ∏r² and height h.
V = ∏r²h
Step 4 :
A sphere of the same radius that the cylinder has, we will find that the sphere can fill two-third of the cylinder.
So, we have
Volume of sphere = 2/3 · Volume of cylinder
Volume of sphere = 2/3 · ∏r²h
(Here, radius and height of the sphere and radius and height of the cylinder are equal)
Step 5 :
A sphere always has a height which is equal to twice the radius.
So, substitute 2r for h.
Volume of sphere = 2/3 · ∏r²(2r)
Volume of sphere = 4/3 · ∏r³
A cone has a radius of r and a height of 2r. A sphere has a radius of r. Compare the volume of the sphere and cone.
The cone’s volume is one-third of a cylinder with radius r and height 2r. The sphere’s volume is two-third of the volume of this cylinder. So, the sphere’s volume is twice the cone’s volume.
After having gone through the stuff given above, we hope that the students would have understood, how to model the volume of a sphere.
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