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

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?

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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?

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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 $\\frac{4}{3} \\pi R^3$.

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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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