(Segmented) Spherical Area Formula

PhotoEditor_20190813_045733325

(UPDATE: It seems that Tempe^4 or 6.3238095 is NOT the exact Hemisphere Area).

This is not final yet.
Just another alternative for Segmented Spherical Area = (1-CosX) × Tempe^4 × r².

(Tempe = 1/4 C)

Let’s see which one is finally true.

============

Segmented Spherical Area = (half radian : slant height)³ × cone area

AKA

Segmented Spherical Area =(rad/l)³ × Phew.l.r

AKA

Segmented Spherical Area = rad³/l² × Phew.r

Exp:

I.

For angle = 180°
Defined Angle = Angle:2 = 180°:2=90°
r=1
rad = (180/2):90 × 1.5857864 = 1.5857864
l =√{[Sin(180/2)]² + [1-Cos(180/2)]²}= √2
rad= 1.5857864
Sphere Area = [1.5857864³/(√2)²] × 3.17157 × 1 = 6.323809

II.

For angle = 90°
Defined Angle = 90°:2 = 45°
r = 0.7071068
rad = 45/90 × 1.5857864 = 0.7928932
l =√{[Sin(90/2)]² + [1-Cos(90/2)]²}= 0.7653668

Sphere Area = (0.7928932³/0.7653668²) × 3.17157 × 0.7071068 = 1.90837

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