# Cone Volume — One Third of a Cylinder
> **Animation coming** — full Sora 2 prompt below. Video will replace this placeholder.
## Plain English first
A cone has the same base and height as a cylinder. How much of the cylinder does it fill?
Exactly one third.
If you fill a cone with water and pour it into a matching cylinder, you need to do it three times to fill the cylinder. This is not a coincidence or an approximation — it is always exactly ⅓, for any cone. The ⅓ in the formula is not arbitrary: it is a geometric fact that three cones fill one cylinder.
---
## Standard math notation
```
V = ⅓ × π × r² × h
Where:
V = volume of the cone
r = radius of the circular base
h = perpendicular height (from base center to apex)
πr² = area of the base circle
⅓ = because three cones always fill one cylinder of same r and h
Compare to cylinder:
V_cylinder = πr²h
V_cone = ⅓ πr²h = V_cylinder / 3
Total surface area:
T = πrl + πr²
where l = slant height = √(r² + h²)
```
---
## Verbose Python with descriptive names
```python
PI = 3.141592653589793
def compute_cylinder_volume(radius_of_base, height):
return PI * radius_of_base ** 2 * height
def compute_cone_volume(
radius_of_circular_base,
perpendicular_height_from_base_to_apex
):
"""
A cone is exactly one third of the cylinder with the same base and height.
Three cones poured into that cylinder fill it exactly.
"""
volume_of_matching_cylinder = compute_cylinder_volume(
radius_of_base=radius_of_circular_base,
height=perpendicular_height_from_base_to_apex
)
volume_of_cone = volume_of_matching_cylinder / 3
return volume_of_cone
def compute_cone_slant_height(radius_of_base, perpendicular_height):
"""
The slant height l is the distance from base edge to apex along the surface.
It's the hypotenuse of the right triangle formed by r and h.
"""
slant_height = (radius_of_base ** 2 + perpendicular_height ** 2) ** 0.5
return slant_height
def compute_cone_surface_area(radius_of_base, perpendicular_height):
"""
Surface = circular base + curved side (which unfolds into a circular sector).
Curved side area = π × r × slant_height
"""
slant_height = compute_cone_slant_height(radius_of_base, perpendicular_height)
area_of_base_circle = PI * radius_of_base ** 2
area_of_curved_side = PI * radius_of_base * slant_height
total_surface_area = area_of_base_circle + area_of_curved_side
return total_surface_area
# Verify: three cones = one cylinder
r, h = 3, 5
print(f"Cylinder: {compute_cylinder_volume(r, h):.4f}") # 141.3717
print(f"Cone × 3: {compute_cone_volume(r, h) * 3:.4f}") # 141.3717 ✓
```
---
## Sora 2 video prompt
```
8-second animation. Left: a transparent cylinder. Three identical cones appear
in sequence — first poured in, fills ⅓; second poured, fills ⅔; third poured,
fills exactly to the brim. Text: '3 cones = 1 cylinder'. Formula assembles:
V = ⅓ × πr²h. The ⅓ glows. Satisfying pour-and-fill motion, dark background,
earth tone geometry, clean 3D style.
```
---
## Builds on
- [Cylinder Volume — A Circle Grown Tall](/articles/cylinder-volume)
- [Circle Area — Why πr²?](/articles/6156ae8e-6c73-4c17-8916-8259a21cad23)
## See also
- [Sphere Volume — Archimedes' Discovery](/articles/sphere-volume)
- [Pythagorean Theorem](/articles/pythagorean-theorem) — used to find slant height
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)