# Sphere Volume — Archimedes' Discovery
> **Animation coming** — full Sora 2 prompt below. Video will replace this placeholder.
## Plain English first
Archimedes figured this out around 250 BCE and was so proud he asked for it to be carved on his tombstone.
The trick: take a cylinder that just barely fits around a sphere (same radius, height = 2r). Now carve out two cones from that cylinder, one pointing up and one pointing down, both with radius r and height r.
What remains inside the cylinder after removing those two cones — that curved shape — has the same volume as the sphere.
So: sphere = cylinder − 2 cones = πr²(2r) − 2×(⅓πr²r) = 2πr³ − ⅔πr³ = **⅔πr³**... wait, that's ⅔. The full sphere is **4/3 πr³** because each half-sphere is ⅔ of a cylinder of height r, and there are two halves.
The 4/3 is not arbitrary. It falls out of the geometry every time.
---
## Standard math notation
```
V = (4/3) × π × r³
Where:
V = volume of the sphere
r = radius
Archimedes' relationship:
V_sphere = (2/3) × V_circumscribed_cylinder
V_cylinder = πr²(2r) = 2πr³
V_sphere = (2/3)(2πr³) = (4/3)πr³ ✓
Surface area (for reference):
T = 4πr²
(exactly 4 times the area of a great circle — another Archimedes result)
```
---
## Verbose Python with descriptive names
```python
PI = 3.141592653589793
def compute_sphere_volume(radius_of_sphere):
"""
Sphere volume = 4/3 × π × r³
Discovered by Archimedes via the cylinder-minus-two-cones argument.
The circumscribed cylinder has volume 2πr³; the sphere is 2/3 of that.
"""
radius_cubed = radius_of_sphere ** 3
volume_of_sphere = (4 / 3) * PI * radius_cubed
return volume_of_sphere
def compute_sphere_surface_area(radius_of_sphere):
"""
Surface area = 4πr²
Exactly four times the area of the circle with the same radius.
Also discovered by Archimedes.
"""
surface_area = 4 * PI * radius_of_sphere ** 2
return surface_area
def verify_archimedes_relationship(radius):
"""
Confirm: sphere volume = (2/3) × circumscribed cylinder volume.
"""
height_of_cylinder = 2 * radius
volume_of_cylinder = PI * radius**2 * height_of_cylinder
volume_of_sphere = compute_sphere_volume(radius)
ratio = volume_of_sphere / volume_of_cylinder
return ratio # always exactly 2/3
print(compute_sphere_volume(radius_of_sphere=3)) # 113.0973...
print(compute_sphere_surface_area(radius_of_sphere=3)) # 113.0973... (coincidence for r=3!)
print(verify_archimedes_relationship(radius=5)) # 0.6666... = 2/3 ✓
```
---
## Sora 2 video prompt
```
8-second animation. A transparent cylinder appears, height=2r. A sphere
materializes inside it, fitting exactly. Two cones form inside the cylinder
above and below the sphere — one pointing up, one down, both with radius r
and height r. The cones glow and dissolve, leaving only the sphere. Text:
'Cylinder − 2 cones = Sphere'. Formula: V = 4/3 πr³. Dark background,
glowing geometry, elegant 3D rotation.
```
---
## Builds on
- [Cone Volume — One Third of a Cylinder](/articles/306a7711-f573-481b-86a2-edfe3e7e55b7)
- [Cylinder Volume — A Circle Grown Tall](/articles/7f98e02b-50ce-41a4-828c-c2fb80b30d97)
- [Circle Area — Why πr²?](/articles/6156ae8e-6c73-4c17-8916-8259a21cad23)
## See also
- [Visual Pi Dependencies](/articles/ec3ebdf4-f6cc-47bf-b597-56fc7b53d130)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)