A cone with the same base and the same height as a cylinder has exactly one-third the volume of the cylinder. This is one of the most surprising facts in elementary 3D geometry — the cone tapers smoothly from base to apex, so you might guess it has half the cylinder’s volume (the “average” radius is half the cylinder’s radius), but it is actually one-third. This guide proves the 1/3 factor three different ways: a hands-on classroom demonstration, an integral calculus derivation, and Cavalieri’s principle. Each proof gives a different intuition for why 1/3 is correct.
For a base radius r and height h:
The cone formula is the cylinder formula multiplied by 1/3. That 1/3 is the part everyone wants explained.
This is the proof most teachers use in middle school or early high school. It is empirical, not symbolic, but it leaves no doubt.
You will find: exactly three coneloads fill the cylinder. The cylinder’s volume is three times the cone’s. The cone holds 1/3 of the cylinder.
This demo works regardless of the specific radius or height — as long as the cone and cylinder match. The 1/3 ratio is universal.
For a more rigorous proof, take a cone with base radius r and height h, oriented with apex at the origin pointing up the z-axis. At height z from the apex, the cone’s radius at that level is (by similar triangles):
radius at height z = (r / h) × z = rz/h
The cross-section at height z is a circle of radius rz/h, with area π(rz/h)² = πr²z²/h².
To find the volume, integrate the cross-sectional area from z = 0 (apex) to z = h (base):
V = ∫₀ʰ π r² z² / h² dz
Pull the constants out:
V = (π r² / h²) × ∫₀ʰ z² dz
The integral of z² from 0 to h is z³/3 evaluated at h minus the value at 0, which is h³/3 − 0 = h³/3.
V = (π r² / h²) × (h³ / 3) = π r² h / 3 = (1/3) π r² h.
The 1/3 factor comes from the integral of z² — that is, from the fact that the cone’s radius grows linearly with height, so its cross-sectional AREA grows quadratically (∝ z²), and integrating z² gives z³/3 (the source of the 1/3).
The Italian mathematician Bonaventura Cavalieri (1598-1647) discovered a principle that lets you compare volumes without integration: two solids of equal height have the same volume if their cross-sections at every horizontal level have the same area.
Start with a cube of side h. The cube has volume h³. Now construct three identical pyramids, each with a square base of side h and height h, that together exactly fill the cube. (This is the classic “three pyramids stacked” demonstration.) Since the three pyramids exactly fill the cube, each pyramid has volume h³/3.
This proves the 1/3 factor for square-base pyramids. Now invoke Cavalieri: any pyramid or cone with the same base area and the same height as one of these square pyramids has the same volume. The cross-section at every level matches in area (because cross-sectional area depends only on the linear scaling factor, which is the same at every level for cones and pyramids of identical height and base area).
So: cone volume = (base area)(height)/3. For a circular base of radius r: base area = πr². So V_cone = πr²h/3.
The cone’s “average” radius (averaged from 0 at the apex to r at the base) is r/2. So you might guess the cone is like a cylinder of radius r/2, giving volume π(r/2)²h = πr²h/4.
But that calculation is wrong because the cone’s radius is NOT uniformly r/2 throughout — it is 0 at the apex, slightly above 0 near the apex, and grows linearly to r at the base. Most of the cone’s volume is in the lower portion (where the cross-sections are larger), so the “effective” radius is bigger than the average.
The integral of z² rather than z² being constant is exactly what captures this. The 1/3 comes from integrating the squared linear function — quadratic growth integrates to cubic, and ∫₀ʰ z² dz / h³ = 1/3 is the universal answer for any shape whose cross-section scales as the square of distance from the apex.
In 2D, a “cone” is a triangle. Its area is ½(base)(height) — that’s the 1/2 factor. The triangle is to the rectangle of equal base and height what the cone is to the cylinder.
In 3D, the cone over a 2D base has volume (1/3)(base area)(height). The 1/3 factor.
In 4D, the cone over a 3D base has hypervolume (1/4)(base volume)(height). The 1/4 factor.
The general pattern: a cone in n+1 dimensions has volume (1/(n+1)) × (base measure in n dimensions) × height. The 1/(n+1) factor comes from integrating xⁿ.
So the 1/3 for the 3D cone is part of a family: it’s just the n = 2 case of the general formula. Not arbitrary at all.
An ice cream cone has a base radius of 2.5 cm and a height of 10 cm. What is its volume?
V = (1/3) π r² h = (1/3) × π × 2.5² × 10 = (1/3) × π × 6.25 × 10 = 62.5π/3 ≈ 65.45 cm³.
For comparison, a cylinder with the same radius and height has volume πr²h = π × 6.25 × 10 = 62.5π ≈ 196.35 cm³. The cone is exactly one-third: 196.35 / 3 = 65.45. ✓
Interestingly, the cone’s surface area does NOT have a clean 1/3 relationship to the cylinder’s. The cone’s lateral surface area is πrℓ where ℓ = √(r² + h²) is the slant height. The cylinder’s lateral surface is 2πrh. These are unrelated by a constant factor.
The 1/3 rule applies specifically to volume — a measure of 3D content. Surface area is a 2D measure of the boundary, and it follows different geometric relationships.
The Sphere/Cylinder/Cone Calculator handles all three shapes — select cone, input radius and height, and get the volume + surface area. The Cone Formula page is a dedicated reference for the formulas. For the 3D Pythagorean theorem that gives the slant height ℓ from r and h, see the 3D Pythagorean Theorem Calculator.
What about a frustum (cone with the top cut off)? A frustum’s volume is V = (1/3)πh(R² + Rr + r²), where R and r are the two parallel circular radii and h is the perpendicular distance between them. Compare: it reduces to the cone formula when r = 0 (pointy top) and to a cylinder when R = r.
Does the 1/3 work for oblique cones? Yes. An oblique cone (apex not directly above the center of the base) has the same volume formula V = (1/3)(base area)(height), where height is still the perpendicular distance from the apex to the base plane. The cone is “tilted” but the volume rule is unchanged. This is a direct consequence of Cavalieri’s principle.
Why is sphere volume (4/3)πr³ — does the 1/3 also appear there? Yes! The sphere’s (4/3) is a 3D volume calculation that involves a similar integration, but the factor isn’t isolated as cleanly. The cone’s 1/3 and the sphere’s 4/3 both ultimately trace back to integrating linear or quadratic radial functions. They’re cousins.