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ลงชื่อเข้าใช้ Mathigon

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บัญชีใหม่     รีเซ็ตรหัสผ่าน     

Circles and PiCavalieri

The volume of an oblique cylinder turns out to be exactly the same as that of a right cylinder with the same radius and height. This is due to Cavalieri’s Principle

Cavalieri’s Principle states that if two solids have the same height and the same cross-sectional area at every level, then they both have the same volume.

We can use this fact to derive that the volume of prisms and cylinders is the area of their cross-section multiplied by their height.

, named after the Italian mathematician Bonaventura Cavalieri

Bonaventura Cavalieri (1598 – 1647) was an Italian mathematician and monk. He developed a precursor to infinitesimal calculus, and is remembered for Cavalieri’s principle to find the volume of solids in geometry.

Cavalieri also worked in optics and mechanics, introduced logarithms to Italy, and exchanged many letters with Galileo Galilei.

Timeline

: if two solids have the same cross-sectional area at every height, then they will have the same volume.

Imagine slicing a cylinder into lots of thin disks. We can then slide these disks horizontal to get an oblique cylinder. The volume of the individual discs does not change as you make it oblique, therefore the total volume also remains constant:

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