04 // Interactive Formula Compendium // GEOMETRY-TRIGV_sphere = (4/3)πr³, V_cyl = πr²h, V_cone = (1/3)πr²h
Sphere, Cylinder & Cone Volume & Surface Area — Solid Mensuration Sheet
For radius r and height h, a right cylinder has volume V_cyl = πr²h, an inscribed cone of identical base and height has volume V_cone = ⅓πr²h, and a sphere of radius r has volume V_sphere = 4/3 πr³ and surface area 4πr².
Formula Reference
Main Formula
V_sphere = (4/3)πr³, V_cyl = πr²h, V_cone = (1/3)πr²h
Rearranged Forms
A_sphere = 4πr² (Sphere Surface Area = dV/dr)
A_cyl = 2πr² + 2πrh (Total Cylinder Surface Area)
A_cone = πr² + πr√(r² + h²) (Cone Base + Lateral Area)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| V_cyl | Right Circular Cylinder Volume (πr²h) | m³ |
| r | Radius (r) | m |
| h | Height (h; set h = 4r/3 for equal sphere volume) | m |
Inputs & Parameters
[m]
[m]
Calculated Result (Right Circular Cylinder Volume (πr²h))
V_cyl =282.7433m³
Formula Notes & Background
- Integrate circular cross-sections of radius r along height [0, h] for a cylinder: V = ∫₀ʰ πr² dz = πr²h.
- For a cone, cross-section radius scales linearly as r(z) = r(z/h), so V = ∫₀ʰ πr²(z²/h²) dz = ⅓πr²h.
- By Cavalieri's Principle (Archimedes' tombstone theorem), a hemisphere of radius r has the same cross-sectional area at every height z as a cylinder of radius r and height r minus an inverted cone, yielding V_sphere = 2(πr³ − ⅓πr³) = (4/3)πr³.
Step-by-Step Examples
#1: Compute the volume of a cylindrical storage tank of radius r = 3 m and height h = 10 m.
r = 3 mh = 10 m
V_cyl = π × 3² × 10 = 90π m³
→ V_cyl = 282.743 m³ (and a cone of same dimensions holds 94.248 m³)
#2: Find the volume and surface area of a sphere with radius r = 3 m.
r = 3 m
V = (4/3)π(3³) = 36π; A = 4π(3²) = 36π
→ V = 113.097 m³, A = 113.097 m²
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Frequently Asked Questions
What was Archimedes' famous 1 : 2 : 3 ratio of cone, sphere, and cylinder?+
For a cone, sphere, and cylinder all of radius r and height h = 2r, their volumes are ⅔πr³ : (4/3)πr³ : 2πr³, which stand in the exact integer ratio 1 : 2 : 3.
Why is the surface area of a sphere 4πr² equal to the derivative of its volume (4/3)πr³?+
Adding an infinitesimal spherical shell of thickness dr increases the volume by dV ≈ A(r)·dr, so dV/dr = 4πr².
What is the slant height ℓ of a right circular cone?+
By the Pythagorean theorem on the radius r and vertical height h, the slant height is ℓ = √(r² + h²), and the lateral surface area is πrℓ.