Question

Describe how the formulas for a cone, cylinder and pyramid are derived. What are the similarities between the volume of a cone and pyramid?

Answer 1

for example, the volume V of a cone whose radius is \(r\), and height is \(h\), is:
\[V = \pi {r^2}h\]

Answer 2

whereas the volume of a cone, with the same radius and height, is:
\[V = \frac{1}{3}\pi {r^2}h\]

Answer 3

and we can see that the ratio between such volume is \(1/3\), namely:
\[\frac{{{V_{CONE}}}}{{{V_{CYLINDER}}}} = \frac{1}{3}\]
provided that both cone and cylinder have the same radius and height

Answer 4

Wait what? You said volume of a cone twice.

Answer 5

I'm saying that if we consider a cone and a cylinder with the same radius of base and the same height, then the volume of cylinder is three time the volume of the cone

Answer 6

What I mean is you gave me the volume of a cone then said whereas the volume of a cone then you gave me a different formula. Did you make a typo?

Answer 7

no, no, I gave the right formulas

Answer 8

So the volume of a cone has two formulas?

Answer 9

the volume V of a cone, is:
\[{V_{CONE}} = \frac{1}{3}\pi {r^2}h\]

Answer 10

whereas the volume of a cylinder, is:
\[{V_{CYLINDER}} = \pi {r^2}h\]

Answer 11

Okay that's where you confused me, thank you. Because you said volume of a cone gave me the formula then said volume of a cone and gave me a diff formula so you made a typo. It's okay, thought I'd clarify bc it confused me ahah

Answer 12

sorry you are right, in the first reply please delete the word "cone" and correct it with the word "cylinder"

Answer 13

Okay, please keep explaining things ^~^ I'm taking notes as you go

Answer 14

I redo, here the corrected first reply:

for example, the volume V of a cylinder whose radius is r , and height is h , is:
\[{V_{CYLINDER}} = \pi {r^2}h\]

Answer 15

here the same second reply:

whereas the volume of a cone, with the same radius and height, is:
\[{V_{CONE}} = \frac{1}{3}\pi {r^2}h\]

Answer 16

finally, here is the same third reply:

and we can see that the ratio between such volume is 1/3 , namely:
\[\frac{{{V_{CONE}}}}{{{V_{CYLINDER}}}} = \frac{1}{3}\]
provided that both cone and cylinder have the same radius and height

Answer 17

Okay. So does that explain how they are kinda derived? I'm not too sure what derived even means haha.

Answer 18

If we know the formula for volume of a cylinder, then I can derive the formula of the volume of a cone with the same radius and the same height, simply dividing by three the formula of the volume of the cylinder

Answer 19

namely:
\[{V_{CONE}} = \frac{{{V_{CYLINDER}}}}{3}\]

Answer 20

Okay (-: How would we derive the formula of a pyramid?

Answer 21

In a similar way, if I consider a prism with an hexagonal base, whose side is \(l\), and height is \(h\), then its volume \(V\), is:
\[{V_{PRISM}} = base\;area \times height\]

Answer 22

next, if we consider a pyramid with hexagonal base with the same side \(l\) and the same height \(h\), then its volume, is:
\[{V_{PYRAMID}} = \frac{{{V_{PRISM}}}}{3}\]

Answer 23

wherein, I repeat, both prism and pyramid have the same height \(h\) and the same side of base \(l\)

Answer 24

Okay, and last question, what are the similarities between the volume of a cone and pyramid? (-:

Answer 25

I think that both formulas have the number \(3\) at the denominator, and the height \(h\) compares, in both formulas, with exponent \(1\)

Answer 26

Thank you so so much, I appreciate your help!! (-:

Answer 27

:)

Answer 28

sorry, I meant this:
I think that both formulas have the number \(3\) at the denominator, and the height h \(appears\), in both formulas, with exponent \(1\)