Wentworth's Plane and Solid Geometry · Book VIII, §§ 838–849 (Props. XXXI–XXXII) · Book 9: Book VIII: The Sphere

128. Volume of a sphere, sector and segment

Goal: Your student can define spherical pyramid, sector, segment and wedge, and prove V = ⅓ S × R = 4/3 πR³, with the formulas for sector and segment.

⏱ About 40 minutes · 7 steps

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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 (§§ 838–843), Proposition XXXI (§§ 844–848) and Proposition XXXII, the volume of a spherical segment (§ 849).

The lesson, step by step

  1. Warm up

    Ask: Surface of a sphere? Volume of a pyramid?

  2. The new idea

    The volume of a sphere is equal to the product of the area of its surface by one third of its radius.

    § 845: V = 4/3 πR³, or ⅙πD³.

    § 848: a spherical sector = ⅓ × zone × R = ⅔πR²H.

    § 849: a segment of one base = ½πr²a + ⅙πa³ (r the radius of the base, a the altitude).

  3. Draw the figure

    Think of the sphere as made of many thin pyramids with their tips at the centre and their bases on the surface, like the seeds of a dandelion clock. Draw one such pyramid.

  4. Walk through the proof

    XXXI: circumscribe a cube, then cut off its corners with more and more tangent planes. Each circumscribed solid is a set of pyramids with altitude R, so its volume is always its surface × ⅓R. Take the limit.

    XXXII: the segment of one base is a sector minus a cone; with r² = a(2R − a) the formula becomes ½πr²a + ⅙πa³. A segment of two bases is the difference of two segments of one base.

  5. 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.

    For Prop. XXXII, it is enough that your student can explain "sector minus cone" and use the final formula.

  6. Exercises

    Compute the volume of a ball of radius 3 in (36π ≈ 113.1 cu in). Then the volume of a cap 1 in high cut from it: r² = 1 × 5 = 5, so ½π × 5 × 1 + ⅙π ≈ 8.38 cu in.

  7. Wrap up

    Ask: If the radius of a ball doubles, what happens to its volume? (8 times.)

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