A cone and cylinder have the same base and height. Why does the cone hold a third rather than half as much? Its axial triangle does occupy half the rectangle, but the area of one slice does not determine volume.
Measure space with cubes
Prerequisites: scale and area, prisms, pyramids, cylinders and cones, sphere and ball.
Volume compares a solid with a unit cube. A 6×4×3 cm rectangular block contains 72 unit cubes of 1 cm³: V=abc. Add component volumes when their interiors do not overlap; shared boundaries add no volume.
A prism has V=Bh, with base area B and perpendicular distance h between base planes. This includes oblique prisms: do not substitute a lateral edge for h. The general pyramid theorem gives V=Bh/3 for the same B and h; the base need not be regular.
One third: a cube check
Join a cube’s center to its vertices. This partitions it into six congruent square pyramids, one per face. A cube of edge 6 cm has volume 216 cm³; each pyramid has 36 cm³. Its base area is 36 cm² and its height only 3 cm. A prism with that same base and height has 108 cm³: 36/108=1/3.
This proves the ratio for the cube model and similar copies. The dissection alone does not prove every base shape and height; a separate general calculation appears below.
Cube: six congruent pyramids
Axial sections: area ≠ volume
For the branch: equal slice areas
A right circular cylinder has V=πr²h; a cone has V=πr²h/3. For r=4 cm, h=3 cm: 48π≈150.80 cm³ versus 16π≈50.27 cm³. Their axial triangle and rectangle have areas 12 and 24 cm². Here the area ratio is 1/2, the volume ratio 1/3: volume depends on all parallel sections.
The ball occupies space; the sphere has area
For R>0 these are different quantities:
V = (4)/(3)π(R)^(3), S = 4π(R)^(2).
At R=3 cm, V=36π cm³ and S=36π cm². The same number 36π is a coincidence in these units; area and volume cannot be compared as like quantities. Sphere area equals the lateral area of its circumscribed cylinder of radius R and height 2R, excluding both bases.
Branch · DeeperOptional · may be skipped
Cavalieri: compare every level
Accept Cavalieri’s principle: solids of equal height with equal cross-sectional areas at every corresponding level have equal volumes. Equal bases alone are insufficient.
At height z above its equator, an upper hemisphere has section area π(R²−z²), 0≤z≤R. Compare a cylinder of radius R and height R with a cone removed. The cone’s apex is at the bottom, z=0, its base at the top, z=R; its section radius is z. The remaining annulus has area πR²−πz² at every level. Hence hemisphere volume is πR³−πR³/3=2πR³/3; doubling gives the ball formula. This uses the already known cone volume.
Branch · With formulasOptional · requires the integral concept
Integrals explain the factors
A pyramid or cone section at distance z from its apex is similar to its base with scale z/h. Its area is B(z/h)², so:
V = ((∫)_(0))^(h)B(((z)/(h)))^(2) dz = (Bh)/(3).
For sphere area parameterize a semicircle as x=R cos t, y=R sin t, 0≤t≤π. The arc element is ds=R dt. Surface-of-revolution area adds strips 2πy ds, yielding S=2πR²∫₀^π sin t dt=4πR². This is a separate area argument, not a silhouette measurement.
Liters and scale
1 cm³=1 mL; 1 L=1 dm³=1000 cm³; 1 m³=1000 L=1,000,000 cm³. Our cylinder holds about 0.15080 L, the cone 0.05027 L. Capacity uses internal dimensions, not a shell’s external diameter.
Scaling all lengths by s>0 multiplies areas by s² and volumes by s³. Doubling a ball’s radius quadruples its area and multiplies volume by eight. Identify the solid, perpendicular height and units before choosing the formula.
Sources and limits
Formulas and conditions: Merzliak and coauthors, 2019, sections 22–23, pp.141–149. AMSI: cube dissection and Cavalieri. OpenStax, Calculus Volume 2, §7.2: surface-of-revolution area. Units: NIST table. Numbers and figures are our own; the ideal model ignores material thickness.