Wentworth's Plane and Solid Geometry · Book VII, Props. XLII–XLIII · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
113. The frustum of a cone
Goal: Your student can find the lateral area and volume of a frustum of a cone and solve the exercises on 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
Propositions XLII–XLIII (§§ 727–730) and Exercises 692–700.
The lesson, step by step
Warm up
Ask: The frustum of a pyramid: what is its volume?
The new idea
The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height.
The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum.
§ 730: V = ⅓πH(R² + r² + Rr).
Draw the figure
Draw a lampshade-shaped frustum with radii R and r, height H and slant height L.
Walk through the proof
XLII: circumscribe the frustum of a regular pyramid: S' = ½(P + p)L always. Take the limit.
XLIII: inscribe the frustum of a regular pyramid: V' = ⅓H(B' + b' + √(B'b')) always. Take the limit.
§ 730: with B = πR² and b = πr², √(Bb) = πRr.
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.
Exercises
Ex. 693. The radius of the base of a circular cone is 8 feet, and the altitude is 10 feet. Find the area of its lateral surface, the area of its total surface, and the volume.
Ex. 695. The slant height of a right circular cone is 2 feet. At what distance from the vertex must the slant height be cut by a plane parallel to the base, in order that the lateral surface may be divided into two equivalent parts?
Wrap up
Ask Ex. 696: what happens to the volume of a cone if the altitude is doubled? the radius? both?
Answers to the book's problems
Ex. 692: the middle section has radius 16.5 in; each frustum has slant height √68.5 ≈ 8.28 in; lower part about 949 sq in, upper about 767 sq in. Ex. 693: slant height √164 ≈ 12.81 ft; lateral about 321.9 sq ft; total about 522.9 sq ft; volume 640π/3 ≈ 670.2 cu ft. Ex. 694: 1 : √5. Ex. 695: √2 ≈ 1.414 ft from the vertex. Ex. 696: 2V; 4V; 8V. Ex. 697: T = 3πR² = ¾πL². Ex. 698: R = √(T/3π), H = R√3, V = ⅓πR³√3. Ex. 699: L = T/(πR) − R, H = √(L² − R²), V = ⅓πR²H. Ex. 700: the base is T − S, so R = √((T − S)/π), L = S/(πR), V = ⅓πR²√(L² − R²).
Worked out for this site from the scan. If a number in your copy differs, trust the book.