Wentworth's Plane and Solid Geometry · Book VII, Exs. 723–735 · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

117. Similar solids

Goal: Your student uses the rule that similar solids are as the cubes of their dimensions to solve problems.

⏱ About 45 minutes · 6 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 calculator (optional)

What's in the book

Page 359: Exercises on similar solids, Ex. 723–735. This ends Book VII.

The lesson, step by step

  1. Warm up

    Ask: If the edges of a solid are doubled, what happens to its surface? its volume?

  2. A trunk

    Ex. 723. The dimensions of a trunk are 4 feet, 3 feet, 2 feet. Find the dimensions of a trunk similar in shape that will hold four times as much.

  3. Cutting a pyramid

    Ex. 725. A pyramid is cut by a plane parallel to the base which passes midway between the vertex and the plane of the base. Compare the volumes of the entire pyramid and the pyramid cut off.

    Ex. 727. The length of one of the lateral edges of a pyramid is 4 meters. How far from the vertex will this edge be cut by a plane parallel to the base, which divides the pyramid into two equivalent parts?

  4. Cones

    Ex. 730. The volumes of two similar cones are 54 cubic feet and 432 cubic feet. The height of the first is 6 feet; what is the height of the other?

  5. Review the Book

    Your student makes a one-page table of the formulas of Book VII (prism, pyramid, frustum, cylinder, cone, prismatoid) from memory, then checks it against the book.

  6. Wrap up

    Check the answers together.

Answers to the book's problems

Ex. 723: multiply by ∛4 ≈ 1.587: about 6.35, 4.76 and 3.17 ft. Ex. 724: (i) √n; (ii) ∛n. Ex. 725: 8 : 1. Ex. 727: 4/∛2 ≈ 3.17 m. Ex. 728: a/∛3 ≈ 0.693a and a∛(2/3) ≈ 0.874a. Ex. 729: if the part at the vertex is the 3, then a∛(3/7) ≈ 0.754a; if it is the 4, a∛(4/7) ≈ 0.830a. Ex. 730: 12 ft. Ex. 731: ∛3 : ∛4, about 0.909 : 1. Ex. 733: multiply H and R by ∛3. Ex. 735: the whole pyramid is 32 ft high; frustum : pyramid = 37 : 64.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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