Wentworth's Plane and Solid Geometry · Book VII, §§ 731–736 (Prop. XLIV) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
114. The prismatoid formula
Goal: Your student can state the prismatoid formula V = ⅙H(B + b + 4M) and use it for prisms, pyramids, frustums, cylinders and cones.
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 calculator (optional)
What's in the book
The prismatoid (§§ 731–732), Proposition XLIV (§ 733), its use for the other solids (§§ 734–736), and Exercise 701.
The lesson, step by step
Warm up
Ask: Give the volume formulas for a prism, a pyramid and a frustum of a pyramid.
The new idea
The volume of a prismatoid is equal to the product of one sixth of its altitude into the sum of its bases and four times its mid-section.
V = ⅙ H (B + b + 4M), where M is the mid-section, halfway between the bases.
Draw the figure
Draw a prismatoid: a rectangle on the bottom, a smaller rectangle turned a little on top, and triangles or trapezoids joining them. Mark the mid-section.
Walk through the proof
XLIV: take a point P in the mid-section and cut the solid into pyramids from P: one on the lower base (⅙ H × B, since its altitude is ½ H), one on the upper base (⅙ H × b), and lateral pyramids. Each lateral pyramid is worth ⅔ H times the part of the mid-section inside it; together they give ⅔ H × M = ⅙ H × 4M.
§§ 735–736: if b = 0 the formula gives ⅓ HB (pyramid); if B = b = M it gives HB (prism).
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 proof is long. Let your student explain the idea of cutting into pyramids from a point of the mid-section, and check §§ 735–736.
Exercises
Ex. 701. Show that the prismatoid formula can be used for finding the volume of the frustum of a cone, for finding the volume of a cone, for finding the volume of a cylinder.
Wrap up
Ask: Why is this one formula so useful? (It works for every solid in the book, and later even the sphere, Ex. 842.)
Answers to the book's problems
Ex. 701: for a cone, b = 0 and M = ¼B, so V = ⅙H(B + B) = ⅓HB. For a cylinder, B = b = M, so V = HB. For a frustum, the mid-section radius is ½(R + r), so 4M = π(R + r)², and ⅙πH(R² + r² + (R + r)²) = ⅓πH(R² + r² + Rr).
Worked out for this site from the scan. If a number in your copy differs, trust the book.