Why is the formula for the volume of a sphere 4 3?

Since the cylinder/cone and hemisphere have the same height, by Cavalieri’s Principle the volumes of the two are equal. The cylinder volume is πR3, the cone is a third that, so the hemisphere volume is 23πR3. Thus the sphere of radius R has volume 43πR3. Integration.

How do you prove the volume of a sphere?

1. Now. according to the volume of a sphere proof.
2. The volume of a sphere = Volume of a cone + Volume of a cone.
3. That is, the volume of a sphere = =πr2h3+πr2h3.
4. The height of the cone = diameter of sphere = 2r.
5. Thus, replacing h = 2r.

How did Archimedes find the volume of a sphere?

Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume.

What does R equal in the volume of a sphere?

πr
Sphere Formulas in terms of radius r: Volume of a sphere: V = (4/3)πr.

Why is the volume of a sphere used?

Volume of a sphere = 4/3 πr3 If you consider a circle and a sphere, both are round. The difference between the two shapes is that a circle is a two-dimensional shape and a sphere is a three-dimensional shape which is the reason that we can measure the Volume and area of a Sphere.

Is Archimedes real?

5 days ago
Archimedes, (born c. 287 bce, Syracuse, Sicily [Italy]—died 212/211 bce, Syracuse), the most famous mathematician and inventor in ancient Greece. Archimedes is especially important for his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder.

What is the relationship of the volume of cylinder to sphere?

What is the relationship between the volume of the sphere and the volume of the cylinder? (Answer: The sphere takes up two-thirds of the volume of the cylinder.)

How do you find the radius of a sphere in real life?

The radius is half the diameter, so use the formula r = D/2. This is identical to the method used for calculating the radius of a circle from its diameter. If you have a sphere with a diameter of 16 cm, find the radius by dividing 16/2 to get 8 cm. If the diameter is 42, then the radius is 21.

What is the volume of the sphere of radius $r$?

Since the cylinder/cone and hemisphere have the same height, by Cavalieri’s Principle the volumes of the two are equal. The cylinder volume is $\\pi R^3$, the cone is a third that, so the hemisphere volume is $\\frac{2}{3} \\pi R^3$. Thus the sphere of radius $R$ has volume $\\frac{4}{3} \\pi R^3$.

How to find the volume of a sphere from a cone?

If you’re willing to accept that you know the volume of a cone is 1/3 that of the cylinder with the same base and height, you can use Cavalieri, comparing a hemisphere to a cylinder with an inscribed cone, to get the volume of the sphere. This diagram (from Wikipedia) illustrates the construction: look here

What is the area of the cross section of a sphere?

For the cylinder/cone system, the area of the cross-section is $\\pi (R^2-y^2)$. It’s the same for the hemisphere cross-section, as you can see by doing the Pythagorean theorem with any vector from the sphere center to a point on the sphere at height y to get the radius of the cross section (which is circular).

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