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.

Modeling the Volume of a 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 = πr²

Step 3 :

Write the formula for volume V of a cylinder with base area B = πr² and height h.

V = πr²h

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 · πr²h

(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 · πr²(2r)

Simplify.

Volume of sphere = 4/3 · πr³ cubic units

Reflect

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.

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