Volume of Prisms, Pyramids, Cones, and Spheres

What Volume Counts, and the Honest Unit

In the last guide you computed surface area — the amount of skin a solid wears. Volume asks a different question: how much space does the solid fill, how much water would it hold? The honest unit is the unit cube, a little 1-by-1-by-1 block, and the volume of any solid is simply how many such cubes you would need to fill it exactly. A box that is 3 units long, 4 wide, and 2 tall holds 3 times 4 times 2 = 24 of those cubes, and that is the whole idea behind every formula in this guide. Areas come in square units, volumes in cubic units; that is why a volume formula always multiplies three lengths and a surface area only two.

Before the formulas, fix the vocabulary. A solid's base is the face you choose to sit it on, and its height is the perpendicular distance straight up from that base to the topmost point or top face — measured along a line _|_ to the base, never along a slanted edge. Keeping height perpendicular is the single most common slip, so we will guard it carefully. With base area written as B and height as h, three of our four solids will turn out to obey one of just two rules: prism-like solids use B times h, and pointed solids use one third of that.

Prisms and Cylinders: Base Times Height

A prism is a solid with two identical parallel faces — the bases — joined by straight sides, so that every horizontal slice is a perfect copy of the base. Picture a stack of identical playing cards: each card is the base shape, and the stack is the prism. If one card has area B and the stack rises to height h, the volume of a prism is just B times h, the area of one card multiplied by how tall the deck stands. A cylinder is the same story with a circular card, so its volume is the circle's area pi r^2 times the height h, giving pi r^2 h.

Pyramids and Cones: Where the One-Third Comes From

A pyramid has one base and a single apex point above it, with the sides tapering to that tip; a cone is the round version, a circular base narrowing to a point. The volume of a pyramid is one third of the matching prism: (1/3) times B times h, and a cone is likewise (1/3) pi r^2 h. The one-third is genuinely surprising — why a third and not, say, a half? The cleanest honest answer is a dissection you can almost do with your hands.

1. Take a cube and pick one corner. From that corner, three square pyramids fan out, each using one of the three faces meeting away from the corner as its base and the chosen corner as its apex.
2. Those three pyramids are identical in shape and they fit together to fill the cube completely, with no gaps and no overlaps. So three equal pyramids make one cube.
3. The cube's volume is B times h (here the base square and the height are equal), and three identical pyramids share it equally, so each pyramid is one third: (1/3) B h. The argument extends to every pyramid and cone by Cavalieri's principle, since stretching or shearing scales base and height in step.

The Sphere: Two-Thirds of Its Box

The sphere is the crown jewel. Its volume is (4/3) pi r^3, and that lonely-looking number hides a beautiful relationship. Stand a sphere of radius r snugly inside the shortest cylinder that contains it — a can whose radius is r and whose height is 2r, the diameter. The cylinder's volume is pi r^2 times 2r = 2 pi r^3. The sphere's volume, (4/3) pi r^3, is exactly two thirds of that cylinder. Archimedes found this so striking that he asked for the sphere-in-a-cylinder to be carved on his tombstone.

How could anyone have known the answer was two-thirds before calculus existed? Through a clever slicing argument, again a cousin of Cavalieri's principle. Compare the half-sphere with a cylinder of radius r and height r out of which a cone has been scooped, point-down. Slice both at any height: the disk cut from the half-sphere turns out to have exactly the same area as the ring cut from the scooped cylinder, level by level all the way up. Equal slices at every level force equal volumes, and the scooped cylinder's volume is the easy difference (cylinder) minus (cone) = pi r^3 minus (1/3) pi r^3 = (2/3) pi r^3 for the half — doubling gives the whole sphere's (4/3) pi r^3.

Slice both solids at height y (0 <= y <= r):

  HALF-SPHERE                 CYLINDER MINUS CONE
  cross-section is a disk     cross-section is a ring (annulus)
  of radius sqrt(r^2 - y^2)    outer radius r, inner radius y

  disk area                   ring area
  = pi (r^2 - y^2)            = pi r^2 - pi y^2
                             = pi (r^2 - y^2)

  SAME at every height  =>  SAME volume  (Cavalieri)
  half-sphere = pi r^3 - (1/3) pi r^3 = (2/3) pi r^3
  whole sphere = (4/3) pi r^3

Frustums, Composites, and How Volume Scales

Real objects are rarely one clean solid, so two more tools finish the toolkit. First, slice the top off a cone or pyramid with a cut parallel to the base and you get a frustum — a bucket or a lampshade shape. Its volume is the big cone you started with minus the little cone you sliced away, both computed with the one-third rule; there is a tidy closed formula, but the honest way to remember it is just big-minus-small. Second, for any composite shape — a silo that is a cylinder topped by a half-sphere, say — find the volume of each clean piece and add, or subtract a piece that has been hollowed out.

Finally, a scaling fact you met as the square-cube law back in the similarity rung deserves a second look here. If you scale a solid by a factor k — double it, so k = 2 — every length grows by k, but area grows by k^2 and volume grows by k^3. Double a sphere's radius and it holds 2^3 = 8 times as much, not twice as much. This single fact explains why a giant cannot simply be a scaled-up human (weight grows like volume, k^3, but bone strength only like cross-section, k^2), why crushed ice melts faster than a block, and why large animals overheat less readily. Volume's stubborn cube is one of geometry's most far-reaching consequences.