MISSION · Measurement
Cylinders, Cones, and Spheres
Three formulas that are really one formula and two fractions
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The idea in one sentence
A cylinder is a circle dragged upwards, a cone is a third of the cylinder around it, and a sphere is two thirds — which is far easier to remember than three separate formulas.
The cylinder
Same logic as a box: area of the base, times the height.
\[V = \pi r^2 h\]The base is a circle, so its area is \(\pi r^2\), and stacking \(h\) of those layers gives the volume. Any prism works this way, whatever the base’s shape.
Its surface is a circle at each end plus a rectangle wrapped round the side. Unroll that side and its width is the circle’s circumference:
\[SA = 2\pi r^2 + 2\pi r h\]The cone and the sphere
\[V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \frac{4}{3}\pi r^3\]The cone’s third is worth testing with water: fill a cone and pour it into a cylinder of the same radius and height, and it takes exactly three coneloads to fill. The same third applies to every pyramid — a pyramid is a third of the prism it sits inside, whatever the base.
The sphere’s formula looks unrelated until you compare it to the cylinder that just encloses it. That cylinder has height \(2r\), so its volume is \(2\pi r^3\) — and the sphere is exactly two thirds of it. Archimedes proved this, considered it his finest result, and asked for the sphere-in-cylinder diagram on his tombstone.
So for a cylinder, cone, and sphere all of radius \(r\) with height \(2r\):
\[\text{cone} : \text{sphere} : \text{cylinder} = 1 : 2 : 3\]One relationship instead of three formulas.
Radius versus height
The radius is squared or cubed; the height isn’t. Doubling a cylinder’s height doubles its volume. Doubling its radius quadruples it.
That’s why a short wide glass holds more than a tall narrow one of the same proportions, and why pizza sizing rewards the larger diameter so heavily. When a quantity appears squared in a formula, it dominates.
Common mix-ups
- Using diameter instead of radius. Halve it first — the error is a factor of 4 or 8.
- Forgetting the third on a cone or pyramid.
- Cubing in the sphere formula, squaring elsewhere. The sphere has no separate height, so \(r\) does all three dimensions of work.
- Mixing slant height with vertical height on a cone. Volume needs the perpendicular height; the slant is for surface area.
Check yourself
A cylinder and a cone both have radius 3 and height 10. Find both volumes, and say what a sphere of radius 3 would hold.
Cylinder: \(\pi(9)(10) \approx 282.7\). Cone: a third of that, \(\approx 94.2\). Sphere: \(\frac{4}{3}\pi(27) \approx 113.1\).
LEVEL 3 · REFLECT
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