You can grasp the volume and surface area of solids just by stacking coins.
One coin, a circle of area πr², becomes a cylinder of volume πr²h once you stack it to height h; a cone is a third of that and a sphere two thirds when the height equals the diameter, a lovely set of ratios.
Unrolling the curved side into a rectangle also makes the surface-area formulas easy to see.
Here you can change the radius and height with sliders and switch among cylinder, cone, and sphere to see how the volumes compare as 3 : 1 : 2.
Intuition — start by stacking coins
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Solid figures — switch among cylinder, cone, sphere
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🪙 Stack the Coins
①Area of one coin (circle) = πr²
②Stack h coins → cylinder → V = πr²h
③A cone is a stack of progressively smaller coins → 1/3 of the cylinder
④A sphere fits inside a cylinder, taking 2/3 of it (when h = 2r)
Volume and Surface Area of a Prism/Cylinder
Prism Volume
V = Sh = base area × height
Holds for any base shape — base area × height
Cylinder Volume
V = πr²h
A prism whose base is a circle of radius r
Cylinder Surface Area
S = 2πr² + 2πrh = 2πr(r + h)
Two end circles + side (unrolls to a rectangle: width 2πr, height h)
🔍 Unroll the Side
①Cut the cylinder's side and flatten it — you get a rectangle
②Width = circumference = 2πr
③Height = h
④Lateral area = 2πrh
Cone Volume — why 1/3?
Cone Volume
V = 13Sh
Exactly 1/3 of a prism with the same base and height
Cone Volume (formula)
V = 13πr²h
1/3 of πr²h
Cone Lateral Area
S = πrl
l is the slant length. Unrolling gives a sector
💡 Verify by Experiment
①Use a cone-shaped cup with the same base and height as a cylinder cup
②Fill the cone with water and pour into the cylinder — exactly 3 cone-fills
③That confirms the 1/3 intuition without calculus!
Sphere Volume and Surface Area
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Volume comparison — cylinder vs cone vs sphere with same r, h
Sphere Volume
V = 43πr³
Sphere of radius r — 2/3 of a cylinder of height 2r
Sphere Surface Area
S = 4πr²
Equal to four circles of the same radius
🏀 Sphere = 2/3 of the Cylinder
①Inscribe a sphere in a cylinder of height = diameter
②Cylinder V = πr² × 2r = 2πr³
③Sphere V = (4/3)πr³ = 2/3 of the cylinder
④A beautiful result Archimedes discovered!
Work It Out
Example 1
Find the volume of a cylinder with base radius 3 and height 5. (Leave π as is.)
1
Volume of a prism/cylinder = (base area) × (height) = πr² × h.
volume = π × 3² × 5
2
Compute.
= 9π × 5 = 45π
▸ 45π
A cylinder volume multiplies base area by height.
Example 2
Find the volume of a square pyramid with base side 6 and height 10.
1
Volume of a pyramid = (1/3) × (base area) × (height). Base area = 6² = 36.
volume = (1/3) × 36 × 10
2
Compute.
= (1/3) × 360 = 120
▸ 120
A pyramid is one-third of the prism with the same base and height.