Sunday, January 22, 2017

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Circle:  Relationship between Area and Circumference

We know that π is a constant (π ≈ 355/113, but more accurately, π ≈ 3.141592654).  And:

Circumference of a Circle:  C  = 2 * π * r
Area of a Circle: A = π * r^2

Observe that:
C = 2 * π * r
π = C / (2 * r)

And:
A = π * r^2
π = A / r^2

Hence:
C / (2 * r) = A / r^2
(2 * r) / C = r^2 / A
A * 2 * r = r^2 *C
A = C * r / 2

Sphere:  Relationship between Area and Circumference

Volume of a Sphere:  V = 4/3 * r^3 * π
Surface Area of a Sphere:  S = 4 * π * r^2

Solving for π:

V = 4/3 * r^3 * π
3 * V = 4 * r^3 * π
π = (3 * V) / (4 * r^3)

And:

S = 4 * π * r^2
π = S / (4 * r^2)

Then:
(3 * V) / (4 * r^3) = S / (4 * r^2)

Multiply both sides by 4 * r^2:
S = 3 * V / r


Equilateral Triangle:  Relationship between Perimeter and Area


Let a (small a) be the length’s side.   Then the area of the triangle:

A = 2 * (1/2 * a/2 * √3/2 * a) = a^2 * √3/4

With the perimeter:  P = 3 * a,

P = 3 * a
P^2 = 9 * a^2
a^2 = P^2 / 9

And

A = a^2 * √3 / 4
a^2 = 4 * A / √3

P^2 / 9 = 4 * A / √3
A = P^2 * √3 / 36

To summarize:

Circle:  Area and Circumference:  A = C * r / 2
Sphere:   Volume and Surface Area:  S = 3 * V / r
Equilateral Triangle:  Area and Perimeter:  A = P^2 * √3 / 36

The next blog will cover Platonic solids.  At least that’s the plan.  Have a great rest of the weekend.  

Eddie

This blog is property of Edward Shore, 2017.

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