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.