Wentworth's Plane and Solid Geometry · Book IV, Exs. 404–423 · Book 5: Book IV: Areas of Polygons

65. Problems of computation: areas (1)

Goal: Your student computes areas of triangles from their sides and solves numerical area problems.

⏱ About 45 minutes · 5 steps

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You'll need

  • The book, open to these pages
  • Pencil and paper
  • A calculator (optional)

What's in the book

Pages 208–209: worked formulas for the equilateral triangle, for any triangle in terms of its sides (Ex. 404–406), then numerical exercises 407–423.

The lesson, step by step

  1. Warm up

    Ask: What is the area of a triangle with base 10 and altitude 6?

  2. The formulas

    Read Ex. 404 and 405 together. Ex. 405 gives the area of a triangle from its sides, with s half the perimeter:

    Area = √(s(s − a)(s − b)(s − c)).

    Test it on a 3, 4, 5 triangle: s = 6, area = √(6 × 3 × 2 × 1) = 6.

  3. Practice

    Ex. 407. Find the area of a right triangle, if the length of the hypotenuse is 17 feet and the length of one leg is 8 feet.

    Ex. 419. Find the area of a triangle, if the lengths of the sides are 104 feet, 111 feet, and 175 feet.

  4. More

    Your student does four or five of Ex. 408–423, with a sketch for each.

  5. Wrap up

    Check the answers together.

Answers to the book's problems

Ex. 407: 60 sq ft. Ex. 408: the triangle with the smaller base has 3 times the altitude (ratio 3 : 1). Ex. 409: 9 ft. Ex. 410: 16⅞ sq ft (diagonals 4½ and 7½ ft). Ex. 411: 100 sq ft. Ex. 412: 30 sq ft. Ex. 413 (base b, leg c): (b/4)√(4c² − b²). Ex. 414: 16√3, about 27.71 sq ft. Ex. 415: h²/√3, or h²√3/3. Ex. 417: 150 sq ft. Ex. 418: a²h / (2√(a² − h²)). Ex. 419: 5460 sq ft. Ex. 420: 20 ft. Ex. 421: 4896 sq ft (2436 + 2460, cutting along AC). Ex. 422: 5000 sq ft in both cases. Ex. 423: 40 ft.

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

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