Area and Volume¶
Recall (doc/postdoc). Not a first course.
Job: elementary Euclidean measure; later Lebesgue / Hausdorff; scaling \(k^{n}\) in dim \(n\).
Core¶
| Figure | Measure |
|---|---|
| Rectangle | \(\ell w\) |
| Triangle | \(\frac12 bh\) |
| Trapezoid | \(\frac12(b_1+b_2)h\) |
| Circle | \(\pi r^2\) |
| Prism / cylinder | \(B h\) |
| Pyramid / cone | \(\frac13 B h\) |
| Sphere | \(V=\frac43\pi r^3\), \(A=4\pi r^2\) |
Cavalieri: same cross-sectional area \(\Rightarrow\) same volume.
Pappus: surface/volume of revolution $= $ path length of centroid \(\times\) path distance.
Jacobian: under \(C^1\) change of variables, \(dV\) multiplies by \(|\det Df|\) — see multivariable calculus.
Used later for¶
- Measure theory: Lebesgue measure extends elementary content
- Integrals: area as \(\int\)
- Differential geometry: volume forms
- Probability: normalizing constants
Check¶
Why can two sets of equal elementary area fail to be equidissectable by scissors?
Hilbert's third problem: Dehn invariants obstruct polyhedral equidissection in \(3\)D (e.g. regular tetrahedron vs cube of same volume).