Ray's New Practical Arithmetic · Arts. 248–250 · Unit 25: Mensuration
116. Areas of trapezoids, triangles and irregular figures
Goal: Your child finds the area of a trapezoid, of a triangle (from base and altitude, or from three sides), and of a figure split into triangles.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 320 gives the trapezoid rule (Art. 248) with problems, and starts the triangle (Art. 249). Page 321 has the three-sides rule, triangle problems, and Art. 250 for irregular figures. Answers are printed.
The lesson, step by step
Warm-up
Ask: A rectangle 6 by 4 is cut corner to corner. What is the area of each triangle?
Trapezoid
Multiply half the sum of the parallel sides by the altitude.
A field is in the form of a trapezoid; one of the parallel sides is 25 rd., and the other 19 rd.; the width is 32 rd.: how many acres in the field?
Half of 44 is 22; 22 × 32 = 704 sq. rd. = 4 A. 64 sq. rd.
Triangle
Multiply the base by the altitude, and take half the product.
When only the three sides are known:
1. From half the sum of the three sides take each side separately. 2. Multiply the half-sum and the three remainders together, and extract the square root of the product.
Try the book's 13, 14, 15 triangle: half-sum 21; remainders 8, 7, 6; 21 × 8 × 7 × 6 = 7056; √7056 = 84 sq. ft.
Irregular figures
1. Divide the figure into triangles by diagonals. 2. Find the areas of the triangles, and add them together.
Guided practice
Do these together:
The base of a triangle is 15 ft. and its altitude 12 ft.: what is its area?
The parallel sides of a trapezoid are 2 ft. 2 in. and 2 ft. 11 in.; its altitude is 11 in.: what is its area?
Independent practice and check
Your child works the remaining problems on pages 320–321. The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.