Core answer: annulus area S=π(R²−r²) with R the outer and r the inner radius; a circular segment equals its sector minus the isosceles triangle; spherical-cap curved area S=2πRh where h is the cap height, not a radius; torus volume V=2π²Rr²; ellipsoid volume V=(4/3)πabc.
Formula cheat sheet for five shapes
| Shape | Key inputs | Area / volume |
|---|---|---|
| Annulus | outer R, inner r | S=π(R²−r²) |
| Segment | radius R, angle θ | S=½R²(θ−sinθ), θ in radians |
| Spherical cap | sphere R, height h | S=2πRh; V=πh²(R−h/3) |
| Torus | centerline R, tube r | S=4π²Rr; V=2π²Rr² |
| Ellipsoid | semi-axes a, b, c | V=(4/3)πabc |
Annulus and segment: the flat family
Annulus is simply big disk minus small disk. Problems often give diameters — halve them first. The ring width w=R−r alone cannot produce the area; you need both radii.
Segment is the piece a chord cuts off: compute the sector for central angle θ (½R²θ), then subtract the triangle formed by the two radii and the chord (½R²sinθ). A semicircle is the special case θ=π with S=½πR².
Cap, torus, ellipsoid: the solid family
The spherical cap has a lovely property: 2πRh depends only on sphere radius × cap height, not on where the cap sits — equal-height caps anywhere on the same sphere have equal area.
Torus (donut): a circle of radius r revolved around an axis R away. Volume = disk area πr² × path length 2πR = 2π²Rr² — a direct application of Pappus’s theorem.
Ellipsoid: stretch a sphere’s three axes to a, b, c and (4/3)πr³ becomes (4/3)πabc. Rugby balls and planets (slightly equator-bulged) are ellipsoids.
Number tables
| Annulus (R, r) | Area |
|---|---|
| R=5, r=3 | π(25−9)=50.3 |
| R=10, r=6 | π(100−36)=201 |
| R=10, r=9 | π(100−81)=59.7 |
| Cap (R=10) | Curved area |
|---|---|
| h=2 | 2π×10×2=125.7 |
| h=5 | 314.2 |
| h=10 (hemisphere) | 628.3 |
Example: pavers for a ring walkway
A circular flower bed of radius 3m is ringed by a 1.5m walkway: R=4.5m, r=3m.
S = π(4.5²−3²) = π(20.25−9) = 35.3 m². Pavers 200×100mm run about 50 per m²; with 8% waste: 35.3×50×1.08 ≈ 1,908 pieces — round up a full box.
Example: glass for a cap skylight
A skylight is a cap of a 6m-diameter sphere opening, cap height 1.2m: sphere radius R=(opening radius²+h²)÷2h=(3²+1.2²)÷2.4=4.35m.
Curved area S=2πRh=2π×4.35×1.2=32.8 m²; order insulating glass at 32.8 m² plus 5% cutting waste, about 34.5 m². Mistaking h for R (2π×1.2²=9 m²) underestimates by 3.6× — the classic quoting disaster.
Common mistakes
- Mixing diameters and radii: annulus problems often state diameters — halve before squaring; S=π(D²−d²)÷4 is the diameter version.
- Degrees in the segment formula: θ in ½R²(θ−sinθ) must be radians; convert 60° to π/3 first.
- Taking cap height as opening radius: h is the sagitta; the opening radius a satisfies a²=2Rh−h² — never swap them.
- Treating an ellipsoid as a sphere: unequal semi-axes break (4/3)πr³ — Earth’s equatorial radius exceeds its polar radius by ~21km, a 0.34% volume difference.