Wentworth's Plane and Solid Geometry · Book VIII, §§ 817–828 (Props. XXVI–XXVII) · Book 9: Book VIII: The Sphere
126. Zones and the surface of a sphere
Goal: Your student can define zone and lune, and prove that the surface of a sphere is 4πR² and that a zone is its altitude times the circumference of a great circle.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- A ball (an orange or a rubber ball) and a rubber band or string
- A calculator (optional)
What's in the book
Definitions of zone, lune and spherical degree (§§ 817–821), Propositions XXVI–XXVII (§§ 822–823) and corollaries (§§ 824–828).
The lesson, step by step
Warm up
Ask: Lateral area of a frustum of a cone?
The new idea
The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle.
The surface of a sphere is equivalent to four great circles; that is, to 4πR².
The area of a zone is equal to the product of its altitude by the circumference of a great circle.
Draw the figure
Slice a ball (or orange) with two parallel cuts: the peel between them is a zone. Mark a lune with two half great circles from pole to pole.
For Prop. XXVI, draw a line AB and an axis XY, with M the middle of AB and MO perpendicular to AB, reaching the axis.
Walk through the proof
XXVI: the surface swept by AB is a frustum of a cone: AB × 2π × (radius at M). Similar triangles show AB × MO' = CD × MO, giving CD × 2π MO.
XXVII: inscribe half a regular polygon in the semicircle; every chord is the same distance a from the centre, so the surface it sweeps is AE × 2πa. As the sides increase, a approaches R.
§ 826: the same reasoning gives a zone = altitude × 2πR.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
The striking fact: on a sphere, zones of equal height have equal area, whether near the pole or at the equator.
Exercises
Compute: the surface of a sphere of radius 5 cm (100π ≈ 314 sq cm), and a zone of altitude 2 cm on it (20π ≈ 62.8 sq cm).
Wrap up
Ask: How many great circles' worth of area cover a sphere? (Four.) An orange peel will just about cover four circles of its own size: try it.