Wentworth's Plane and Solid Geometry · Book VII, Exs. 662–675 · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

102. Problems of computation: pyramids

Goal: Your student computes volumes, surfaces and heights of pyramids.

⏱ 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 317: Problems of Computation, Ex. 662–675.

The lesson, step by step

  1. Warm up

    Ask: Volume of a pyramid? Lateral area of a regular pyramid? Area of an equilateral triangle of side s? (¼ s²√3.)

  2. Volumes

    Find the volume in cubic feet of a regular pyramid:

    Ex. 662. When its base is a square, each side measuring 3 feet 4 inches, and its height is 9 feet.

    Ex. 663. When its base is an equilateral triangle, each side measuring 10 feet, and its height is 15 feet.

  3. Surfaces

    Ex. 665. When each side of its square base is 10 feet, and the slant height is 18 feet.

    Ex. 667. When each side of its square base is 32 feet, and the perpendicular height is 72 feet.

    For Ex. 667, first find the slant height from the right triangle made by the height and half a side.

  4. Heights

    Ex. 669. The volume is 20 cubic feet, and the sides of its triangular base are 5 feet, 4 feet, and 3 feet.

    Ex. 670. The base edge of a regular pyramid with a square base measures 40 feet, the lateral edge 101 feet. Find its volume in cubic feet.

  5. Formulas

    If time allows, your student tries Ex. 672 or 674, giving the answer as a formula.

  6. Wrap up

    Check the answers together.

Answers to the book's problems

Ex. 662: 33 1/3 cu ft. Ex. 663: 125√3, about 216.5 cu ft. Ex. 664: 160√3, about 277.1 cu ft. Ex. 665: 360 + 100 = 460 sq ft. Ex. 666: 192 + 16√3, about 219.7 sq ft. Ex. 667: slant height √5440, about 73.76 ft; total about 5744 sq ft. Ex. 668: 6.5 ft. Ex. 669: the base is a right triangle of area 6 sq ft, so the height is 10 ft. Ex. 670: height √9401, about 96.96 ft; volume about 51,711 cu ft. Ex. 671: the apothem of the base is 5 ft, the height √119, about 10.91 ft; volume about 472.4 cu ft. Ex. 672: the slant height is L = (T − a²)/(2a); then V = ⅓a²√(L² − ¼a²). Ex. 674: each face is equilateral; T = a² + a²√3, so a = √(T/(1 + √3)).

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