Question

How are the formulas for the volume of a cylinder, pyramid, and cone derived?

Answer 1

Calculus-based analytic geometry makes the cylinder, pyramid, and cone from "solids of revolution". From there, they can measure the length or radius of the base, and then the height. This is true even if the solid of revolution is along the X axis intead of the Y axis. Then someone "figured out" the formula, just like they did with a cube, and proved it.

Answer 2

Answer with explanation:

Volume is a term used for three dimensional Object.

Volume of any three 3-D object is Calculated by the formula:

=Area of Base × Height

For, example

⇒Volume of Cuboid= Area of base which is in the shape of rectangle × Height of Cuboid

⇒Area of Rectangle =Length (L) × Breadth (B)

= L × B ×H

1. Cylinder

Base of cylinder is in the shape of circle.Let h be the height of cylinder.

⇒Volume of Cylinder= Area of base which is in the shape of a Circle × Height of Cylinder

= π r² h

2.Pyramid

A pyramid is a three dimensional solid, whose base is in the shape of any polygon and other faces are Triangles.

⇒Volume of Pyramid= Area of base which is in the shape of a Polygon × Height of Pyramid

3.Cone

A Cone is a three dimensional solid, which is obtained when you rotate a right triangle along it's base either clockwise or anticlockwise and by joining all the vertex of congruent right triangles,from itself as well as from the top of vertex the shape obtained is Cone

⇒Volume of Cone= Area of base which is in the shape of a Circle × Height of cone

`=(\pi)/(3)r^2h`