Cones, Pyramids: Surface Area and Volume FormulasDate: 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 http://mathforum.org/library/drmath/view/55263.html Volume of a Pyramid http://mathforum.org/library/drmath/view/55041.html 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 simple. 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 http://mathforum.org/library/drmath/view/55082.html 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 http://mathforum.org/dr.math/ |
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