why is the formula of finding the volume of a sphere 4/3 pi r3?

Answer 1

that is because when finding the surface area, it is 4 pi r spuared, right?
so when you find the volume, you divide that formula by three.
just like a cone, the volume of a cone is the area of cricle times height over three.
thats one way of looking at it,
the other way, involves the guy who disovered the formula, a rope, cylinder and sphere. but i'm not sure i can make u understand by typing it.
sorry.

Answer 2

Think of a very thin pyramid with its vertex at the centre of the sphere, and its base at the surface.

Its volume is (1/3 x area of base x height).

Make the sphere out of zillions of these thin pyramids. To sum up their volumes, take out the common factor (1/3 x height), and we get (1/3 x height) x (sum of area of all bases).

Now the height is R, and the sum of the areas of all the bases is just the surface area of the sphere, which is 4 x pi x R x R, so there it is - 1/3 x R x 4 x pi x R x R, just what you said.

Answer 3

Think of a cylinder encasing the sphere with radius r and height 2r. That has volume 2 pi r^3. It's easy to see that a sphere should have volume of about half that.

Being more precise is really difficult without calculus. It's possible -- I think the ancient Greeks had that formula -- but in essence they were doing a bit of calculus without realizing it.

In theory it's a problem in calculus of several variables. But in reality, when you take first year calculus and learn how to perform integrals, ask your instructor and s/he will be easily able to prove the volume of the sphere to you.

Answer 4

The second theorem of Pappus states that:
The volume of a solid of revolution is equal to the generating area times the circumference of the circle described by the centroid of the area.

V = (2pi)(y)(A)

where:

y = is the distance from the centroid to the axis of revolution

A = is the area.

Now, imagine a sphere as a semicircle rotated around its base/diameter(longest chord).

since the centroid(y) is located 4r/3pi from the base.

and A = (pi)(r^2)/2 .....area of a semicircle of radius r

therefore

V = 2pi(4r/3pi)(pi)(r^2)/2

simplifying, you'll get:

V = 4(pi)(r^3)/3

Answer 5

Consider a circle of radius r, centre the origin.
Rotate circle about x axis to obtain volume of revolution.
Equation of circle is x² + y² = r²
V = 2 ∫π.y² dx between limits 0 to r
V = 2π∫(r² - x²).dx
V = 2π(r²x - x³/3) between 0 and r
V = 2π(r³ - r³/3)
V = 2π.(2/3).(r³)
V = (4/3).π.r³

Answer 6

You must know integral calculus to prove this