Wentworth's Plane and Solid Geometry · Book VII, Exs. 712–722 · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

116. Equivalent solids

Goal: Your student changes one solid into another of equal volume and finds the missing dimension.

⏱ About 40 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 358: Exercises on equivalent solids, Ex. 712–722.

The lesson, step by step

  1. Warm up

    Ask: What does "equivalent solids" mean? (Equal volumes.)

  2. Cube into prism

    Ex. 712. A cube each edge of which is 12 inches is transformed into a right prism whose base is a rectangle 16 inches long and 12 inches wide. Find the height of the prism, and the difference between its total area and the total area of the cube.

  3. Cylinders and cones

    Ex. 717. The heights of two equivalent right circular cylinders are in the ratio 4 : 9. If the diameter of the first is 6 feet, what is the diameter of the other?

    Ex. 718. A right circular cylinder 6 feet in diameter is equivalent to a right circular cone 7 feet in diameter. If the height of the cone is 8 feet, what is the height of the cylinder?

  4. Frustum into pyramid

    Ex. 719. The frustum of a regular pyramid 6 feet high has for bases squares 5 feet and 8 feet on a side. Find the height of an equivalent regular pyramid whose base is a square 12 feet on a side.

  5. More

    Your student chooses two more from Ex. 713–722. Ex. 714 and 716 are short.

  6. Wrap up

    Check answers. Ask: in Ex. 712, did changing the shape change the surface? (Yes; same volume, different surface.)

Answers to the book's problems

Ex. 712: height 9 in; prism surface 888 sq in, cube 864 sq in, difference 24 sq in. Ex. 713: (i) bc/(πa); (ii) 3bc/(πa). Ex. 714: 4 ft. Ex. 715: 392π/16 ≈ 76.97 ft. Ex. 716: H = a³/(πR²). Ex. 717: 4 ft. Ex. 718: about 3.63 ft. Ex. 719: frustum 258 cu ft; height 5 3/8 ft. Ex. 720: 5 1/15 ft. Ex. 721: about 1.47 in. Ex. 722: about 2.33 in.

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