The Stack of Coins Idea
Take a neat stack of identical coins standing as a tidy cylinder. Now slide your thumb along the side and shear the stack into a leaning, lopsided tower. Not one coin has changed; you have only nudged each to a new horizontal spot. So the total amount of metal — the volume — is exactly the same as before, even though the leaning solid looks quite different. That stubborn fact, that sliding the layers sideways cannot change the total, is the whole heart of Cavalieri's principle.
Stated cleanly: suppose two solids sit between the same pair of parallel planes — a floor and a ceiling. Slice both with any horizontal plane in between. If at every height the two cross-sections have equal area, then the two solids have equal volume. The shapes of the slices may differ wildly from solid to solid; all that matters is that, level by level, their areas match. It is the three-dimensional cousin of an idea you can also run in the plane: two regions between the same two parallel lines, whose horizontal cross-segments have equal length at every level, enclose equal area.
Why It Is True (and Why It Hints at Calculus)
The intuition is exactly the coin stack. Imagine slicing each solid into a great many ultra-thin horizontal wafers, each so thin that its volume is well approximated by its cross-sectional area times its tiny thickness. If at every height the two solids present equal cross-sectional areas, then their matching wafers have equal volumes, one for one. Add up all the wafers in the first solid, add up all the wafers in the second, and you are summing two lists of equal numbers — so the totals agree. That is the argument in spirit, and it is honest as far as it goes.
But notice the soft spot: a wafer of real thickness is not exactly its area times its thickness — the sides may flare or pinch a little across that thickness. Making the wafers thinner shrinks the error, and the honest way to take the error all the way to zero is a limit. That limit is precisely the integral: the volume is the integral of the cross-sectional area A(h) as the height h sweeps from floor to ceiling. Cavalieri worked in the 1630s, before calculus had a rigorous footing, and his principle is best seen as a beautiful, correct shadow of integration cast a generation early. We will use it as a working tool here and leave the full epsilon-and-delta justification to the calculus rung above.
Cashing It In: Why the Pyramid Is One-Third
The previous guide gave you the volume of a pyramid as one-third of base times height, and of a prism as base times height. Cavalieri is what lets the slanted, oblique pyramid share the formula of the upright one. Slice an oblique pyramid and a right pyramid that have congruent bases and the same height by a plane at height h. By similarity, each cross-section is a scaled copy of the base, shrunk by the same ratio that depends only on h — so at every level the two pyramids show cross-sections of equal area. Cavalieri then declares their volumes equal: leaning the pyramid over changes nothing.
The same slicing explains the one-third itself, not just the obliqueness. A classic dissection cuts a triangular prism into exactly three pyramids of equal volume; Cavalieri is the tool that proves those three pieces really are equal in volume, since you can match their cross-sections level by level. Three equal pyramids fill one prism, so each pyramid is one-third of the prism on the same base and height. The mysterious 1/3 is not a number pulled from a hat — it is three slices of one prism, and Cavalieri is the referee certifying the split is fair.
The Showpiece: The Volume of a Sphere
Here is the result that makes everyone fall in love with slicing. We want the volume of a hemisphere of radius r. The trick is to find a second, easier solid whose slices secretly match the hemisphere's, then read off the answer. Stand the hemisphere flat-face down. Beside it, stand a cylinder of radius r and height r, and from that cylinder scoop out a cone that has its point at the bottom centre and its open mouth as the top rim. We will slice all of this at a height h above the floor and compare areas.
Slice the hemisphere at height h. By the Pythagorean theorem the disk there has radius sqrt(r^2 - h^2), so its area is pi times (r^2 - h^2). Now slice the cylinder-minus-cone at the same height h. The cylinder gives a full disk of area pi times r^2. The cone, whose radius grows from 0 at the bottom to r at the top, has radius exactly h at height h, so the hole punched through is a disk of area pi times h^2. The leftover ring has area pi times r^2 minus pi times h^2, which is pi times (r^2 - h^2). The two areas are identical at every height — so by Cavalieri the two solids have equal volume.
At height h, both cross-sections have area pi (r^2 - h^2):
hemisphere slice (a disk) cylinder-minus-cone slice (a ring)
radius sqrt(r^2 - h^2) outer radius r, inner radius h
area = pi (r^2 - h^2) area = pi r^2 - pi h^2 = pi (r^2 - h^2)
Volume(hemisphere) = Volume(cylinder) - Volume(cone)
= pi r^2 * r - (1/3) pi r^2 * r
= (2/3) pi r^3
Whole sphere = two hemispheres = (4/3) pi r^3Now we just read the easy solid. The cylinder has volume pi times r^2 times r, and the scooped cone, by the one-third rule from the last guide, has volume one-third of that. Subtracting leaves two-thirds of pi times r^3 for the hemisphere, and doubling gives the famous (4/3) times pi times r^3 for the whole sphere. No calculus was invoked — only a Pythagorean radius, the area of a circle, the cone's one-third, and the single act of faith that matching slices mean matching volume. That is Cavalieri's principle earning its keep.
How Volume Scales: The Square-Cube Law
Slicing also makes a scaling fact obvious. If you blow up a solid by a factor k in every direction — twice as wide, twice as tall, twice as deep — then each horizontal slice becomes a similar figure scaled by k, so every cross-sectional area is multiplied by k^2. Stack up slices whose areas all grew by k^2, across a height that also grew by k, and the total volume grows by k^2 times k, that is k^3. Lengths scale by k, areas by k^2, volumes by k^3. This is the square-cube law, and it falls right out of the same level-by-level bookkeeping.
The consequences are vivid and a little ruthless. Double a statue's height and you do not double its weight — you multiply it by 2^3 = 8, while its skin (a surface area) grows only by 2^2 = 4. That mismatch is why an ant can shrug off a fall and an elephant cannot, why crushed ice cools a drink faster than one big block, and why a giant the shape of a person could not stand: bones whose cross-section grew by k^2 would have to carry a weight that grew by k^3. The square-cube law is not a quirk of biology; it is geometry insisting that area and volume grow at different rates.
Honest Limits and a Last Look
Cavalieri's principle is powerful, but it has fine print worth stating plainly. The slices must be cut by genuinely parallel planes, and the two solids must sit between the same two of them — a floor and ceiling shared, not merely similar. The matching condition is on area, so you may freely trade a triangular slice for a circular one of equal area, but you may not relax the requirement that they agree at every single height; one mismatched level and the conclusion is void. And the principle compares two solids — it does not, on its own, hand you a number. You still need one easy reference solid whose volume you already trust, which is exactly why the sphere argument leaned on a cylinder and a cone.
Seen from the top of this rung, every volume formula you met is one family. The prism and cylinder are stacks of unchanging slices; the pyramid and cone are stacks of slices shrinking by similarity to a point; the sphere is a stack of disks whose radius bends according to Pythagoras. Cavalieri is the thread running through them — the recognition that a volume is nothing more than its slices, summed. Carry that picture up into calculus and the informal sum of wafers becomes the integral of A(h), the same idea made rigorous. You have been doing baby integration all along, and you did it with nothing fancier than the area of a circle and a steady hand on the knife.