Area and Volume of Solids

Surface area vs volume

A solid is a three-dimensional shape. Two numbers describe how big it is.

• Surface area (SA) is the total area of its outside, like the wrapping paper needed to cover it. It is measured in square units, such as cm².
• Volume (V) is the space inside, like the water it holds. It is measured in cubic units, such as cm³.

The height h of a solid is always measured straight up, at right angles to the base. Answers “in terms of π” keep π as a symbol, which is exact.

Prisms and cylinders

A prism has two matching, parallel bases joined by flat sides. A cylinder is the same idea with circles for bases. For both, stack copies of the base up to height h:

• Volume: V = Bh, where B is the area of the base. For a cylinder, V = πr²h.
• Surface area: two bases plus the sides. Unroll the sides and you get a rectangle whose width is the base’s perimeter P, so SA = 2B + Ph. For a cylinder, SA = 2πr² + 2πrh.

Example: the box in the figure has V = 4 · 3 · 5 = 60 and SA = 2(4·3 + 4·5 + 3·5) = 94. The cylinder has V = π · 9 · 5 = 45π and SA = 18π + 30π = 48π.

Pyramids and cones: the ⅓ factor

A pyramid rises from a polygon base to a single point, the apex. A cone does the same from a circle. Each fills exactly one third of the prism or cylinder with the same base and height:

• Pyramid: V = ⅓Bh
• Cone: V = ⅓πr²h

For surface area you need the slant height l, measured along the outside. On a cone, the radius, height and slant height make a right triangle, so l = √(r² + h²). The curved side of a cone has area πrl, so its total surface area is πr² + πrl. A regular pyramid’s triangular faces add up to ½Pl, half the base perimeter times the slant height.

Example: the cone in the figure has r = 3 and h = 4, so l = 5. Its volume is ⅓ · π · 9 · 4 = 12π, and its total surface area is 9π + 15π = 24π.

Spheres

A sphere is every point in space at distance r from a center. A great circle is the biggest circle you can draw on it, with the same center and radius.

• Volume: V = (4/3)πr³
• Surface area: SA = 4πr², exactly four great circles.

Archimedes found the neatest way to remember these. Put the sphere inside the smallest cylinder that holds it, so the cylinder’s height is 2r. The cylinder’s volume is πr² · 2r = 2πr³, and the sphere is two thirds of that. Its surface, counting the top and bottom, is 6πr², and the sphere’s surface is two thirds of that too. With a cone of the same base and height, the three volumes are in the ratio 1 : 2 : 3.

Cavalieri’s principle

Cavalieri’s principle: if two solids have the same height, and every slice parallel to the base at the same level has the same area in both, then the solids have the same volume.

Think of a neat stack of coins, then push it so it leans. Every coin is the same, so the amount of metal is the same. That is why an oblique (slanted) prism or cylinder still has V = Bh, with h measured straight up, not along the slant. It is also why every pyramid with the same base area and height has the same volume, whatever the base’s shape.

Scaling: k, k² and k³

Two solids are similar when one is a scaled copy of the other. If every length is multiplied by the scale factor k:

• lengths (edges, radii, heights) are multiplied by k;
• areas (including surface area) are multiplied by k²;
• volumes are multiplied by k³.

Areas have two length dimensions and volumes have three. Example: a model car at 1 : 10 scale needs 1/100 of the real car’s paint, and holds 1/1000 of its volume.

Common mistakes

• Using the slant height for volume. Volume formulas use the straight-up height h; the slant height l is only for surface area.
• Forgetting the ⅓. Pyramids and cones are one third of the matching prism or cylinder.
• Using the diameter as the radius. Halve the diameter first.
• Scaling volume by k. Doubling a sphere’s radius multiplies its volume by 8, not 2.

A short history: Euclid, Archimedes, Zu Geng and Cavalieri

Euclid’s Elements (around 300 BCE) proves the ⅓ facts in Book XII. Proposition 7 shows that a prism with a triangular base splits into three equal pyramids, so any pyramid is one third of the prism with the same base and height. Proposition 10 shows that any cone is one third of the cylinder with the same base and height. Proposition 18 shows that spheres are to one another “in triplicate ratio” of their diameters: volume grows like d³. Archimedes later wrote that Eudoxus was the first to prove the pyramid and cone results.

Archimedes (about 287–212 BCE), in Book I of On the Sphere and Cylinder, proved that a sphere’s surface is four times its great circle (Proposition 33) and that its volume is four times the cone whose base is a great circle and whose height is the radius (Proposition 34). A corollary gives the cylinder result: the cylinder around a sphere is 3/2 of it, in volume and in surface. He asked for a sphere and cylinder to be carved on his tomb. In 75 BCE the Roman Cicero, serving in Sicily, found an overgrown tomb with exactly that figure.

In China, Liu Hui, in his commentary on The Nine Chapters on the Mathematical Art (263 CE), showed that the old rule for a sphere’s volume was wrong, but he could not find the right one. Zu Geng (about 450–520), the son of the astronomer Zu Chongzhi, finished the job. His proof, preserved in Li Chunfeng’s 7th-century commentary, rests on the idea that solids whose slices at equal heights match have equal volumes. Today this is called the principle of Liu Hui and Zu Geng.

In Italy, Bonaventura Cavalieri published the same idea, with his “method of indivisibles,” in Geometria indivisibilibus continuorum (1635), more than a thousand years later.

Sources

• Euclid, Elements, ed. D. E. Joyce, Clark University: Book XII, Proposition 7 (a prism splits into three equal pyramids); Proposition 10 (a cone is a third of its cylinder); Proposition 18 (spheres are in triplicate ratio of their diameters).
• Archimedes, On the Sphere and Cylinder, Book I, Propositions 33–34 and corollary, in T. L. Heath, The Works of Archimedes (Cambridge, 1897), Internet Archive scan.
• MacTutor History of Mathematics, University of St Andrews: Archimedes of Syracuse (biography, with Cicero’s account of the tomb); Zu Geng (biography, with his sphere proof and the principle of Liu Hui and Zu Geng); Bonaventura Cavalieri (biography).

Credits: all diagrams on this page are original SVG drawings made for Jugami.