**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.