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
appreciated.

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.

SwimDuck
```

```
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^
2.sin(theta).d(theta)

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

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!  http://mathforum.org/dr.math/
```
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
Math Forum Home || Math Library || Quick Reference || Math Forum Search