# 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
```python
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?](/articles/6156ae8e-6c73-4c17-8916-8259a21cad23)
- [Rectangle Area — Counting the Grid](/articles/66820765-0ec9-4197-8cfa-55bb81c2bf6d)
## See also
- [Cone Volume — One Third of a Cylinder](/articles/cone-volume)
- [Sphere Volume — Archimedes' Discovery](/articles/sphere-volume)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)