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Cones, Pyramids: Surface Area and Volume Formulas

Date: 03/11/2003 at 18:32:29
From: Aditi
Subject: Derivations of formulas

How do you get the formula for surface area and volume in the easiest 
way for cones and pyramids?

I know the equations; I just do not understand how to get them.

Date: 03/11/2003 at 22:55:44
From: Doctor Peterson
Subject: Re: Derivations of formulas

Hi, Aditi.

Both formulas are rather easy to remember in this form:

    V = Bh/3

    S = Ps/2

The volume formula is one of the hardest to actually derive; you can 
read about that here if you want:

   Volume of a Cone 

   Volume of a Pyramid 

But it is easy to understand: the volume is 1/3 the volume of the 
cylinder or pyramid of the same height with the same base. Since the 
latter is Bh, the product of the base area and the height, this is 

The area formula (I've given the lateral area only) for a regular cone 
or pyramid is easier to prove. The lateral surface of a pyramid is 
composed of triangles, all of which have the same height, the slant 
height s. The height of one of these is bs/2, where b is its base; all 
together, the bases of the triangles add up to P, the perimeter of the 
base, so the area is Ps/2. A cone takes the same formula (where the 
perimeter is called the circumference).

An alternative derivation of the surface area of a cone is given here:

   Lateral Surface of a Cone 

Notice the similarity of the formulas: each is something about the 
base as a whole, times a measurement away from the base, divided by 
the dimension of the quantity being calculated. (Volumes are three-
dimensional; areas are two-dimensional.) And the vertical height used 
for the volume is the distance of the apex from the base whose area 
it multiplies, while the slant height used for the surface area is 
the distance of the apex from the perimeter of the base, whose length 
it is multiplied by.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Higher-Dimensional Geometry

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