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

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
• Circle Area — Why πr²?

See also

• Sphere Volume — Archimedes' Discovery
• Pythagorean Theorem — used to find slant height
• Math Foundations — Visual Table of Contents