The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Derivation of Geometric Formulas

Date: 5/29/96 at 18:35:16
From: Anonymous
Subject: Derivation of Geometric Formulas

Could you please answer these questions? Your help would be very much 

How were the formulas for surface area, total surface area, and volume 
of a sphere derived? How were the formulas for the volume of a pyramid 
and cone derived?

Please give an explanation of the answers to the above questions and 
include equations if there are any. Thank you.


Date: 5/31/96 at 11:45:38
From: Doctor Anthony
Subject: Re:Derivation of Geometric Formulas

In the case of the surface area of a sphere, consider an elementary 
element of arc at position (a,theta) (in polar coordinates) rotated 
through 360 degrees about the x axis, thereby forming a hoop of radius 
a.sin(theta) and surface area a.d(theta).2.pi.a.sin(theta) = 2.pi.a^

To get the total surface area we integrate this between theta = 0 and 

Surface area = 2.pi.a^2.INT[sin(theta).d(theta)] between 0 and pi.
             = 2.pi.a^2.[-cos(theta)] between 0 and pi
             = -2.pi.a^2[-1 - 1]
             = 4.pi.a^2

For volume consider the circle x^2 + y^2 = a^2, and take a slice of 
thickness dx, and radius y (= sqrt(a^2-x^2)).  Rotate this about the 
x axis to get a disk of volume pi.y^2.dx.  Integrate between 
x = -a and +a:

Volume = pi.INT[(a^2 - x^2).dx]
       = pi.[a^2.x - (1/3)x^3]  between -a and +a
       = pi.[a^3 - (1/3)a^3 + a^3 - (1/3)a^3]
       = pi.[2a^3 - (2/3)a^3]
       = pi.[(4/3)a^3]

For volume of cone of height h and base radius a, consider the line 
y = (a/h)x. By rotating this about the x axis you will generate the 
cone. The volume of an elementary disk of thickness dx and radius y is 
pi.y^2.dx = pi.(a^2/h^2)x^2.dx  Integrate between 0 and h:

Volume = pi.INT[(a^2/h^2)x^3/3]  between 0 and h
       = pi.(a^2/h^2)(h^3/3)
       = (1/3).pi.a^2.h
In the case of a pyramid, you take slices, parallel to the base, of 
thickness dx and area calculated by ratio of similar rectangles or 
square with the shape of the base.  You get the same formula as for 
the cone, namely(1/3).area of base times perpendicular height. 

-Doctor Anthony,  The Math Forum
 Check out our web site!   
Associated Topics:
College Calculus
College Higher-Dimensional Geometry
High School Calculus
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.