Cylinder Volume — A Circle Grown Tall

Animation coming — full Sora 2 prompt below. Video will replace this placeholder.

Plain English first

A cylinder is just a circle that has been stretched upward.

Start with a circle of radius r at the bottom. That circle has area πr². Now stack an identical circle on top of it, and another, and another — h times. Each layer adds πr² of volume. Stack them all and the total is πr² × h.

Volume = (area of one circular layer) × (number of layers stacked).

This is the same logic as a rectangular box (V = l×w×h), except the base shape is a circle instead of a rectangle.

Standard math notation

V = π × r² × h

Where:
  V = volume (cubic units)
  r = radius of the circular base
  h = height (how tall the cylinder is)
  πr² = area of one circular layer

Total surface area (for reference):
  T = 2πr² + 2πrh
    = 2πr(r + h)
  (two circular caps + one rectangle rolled into the side)

Verbose Python with descriptive names

PI = 3.141592653589793

def compute_circle_area(radius_of_circle):
    return PI * radius_of_circle ** 2

def compute_cylinder_volume(
    radius_of_circular_base,
    height_of_cylinder
):
    """
    A cylinder is a circle extended upward.
    Volume = area of one circular layer × number of layers stacked.
    """
    area_of_one_circular_layer  = compute_circle_area(radius_of_circular_base)
    number_of_layers_stacked    = height_of_cylinder
    total_volume                = area_of_one_circular_layer * number_of_layers_stacked
    return total_volume

def compute_cylinder_surface_area(
    radius_of_circular_base,
    height_of_cylinder
):
    """
    Surface = two circular caps (top and bottom) + curved side wall.
    The side wall, if unrolled, is a rectangle: width = circumference, height = h.
    """
    area_of_one_cap          = compute_circle_area(radius_of_circular_base)
    area_of_both_caps        = 2 * area_of_one_cap
    circumference_of_base    = 2 * PI * radius_of_circular_base
    area_of_side_wall        = circumference_of_base * height_of_cylinder
    total_surface_area       = area_of_both_caps + area_of_side_wall
    return total_surface_area

volume = compute_cylinder_volume(
    radius_of_circular_base=3,
    height_of_cylinder=5
)
print(f"Volume: {volume:.4f}")  # 141.3717

surface = compute_cylinder_surface_area(
    radius_of_circular_base=3,
    height_of_cylinder=5
)
print(f"Surface area: {surface:.4f}")  # 150.7964

Sora 2 video prompt

8-second animation. A glowing circle labeled A=πr² sits at the bottom of
frame. It begins extruding upward — each layer a translucent glowing disk,
stacking h times. The cylinder builds bottom to top with a visible grid
texture. Formula assembles onscreen: V = πr² × h. Final cylinder has slight
3D rotation to show depth. Dark background, warm earth tone layers, technically
clean style.

Builds on

• Circle Area — Why πr²?
• Rectangle Area — Counting the Grid

See also

• Cone Volume — One Third of a Cylinder
• Sphere Volume — Archimedes' Discovery
• Math Foundations — Visual Table of Contents