Wentworth's Plane and Solid Geometry · Book IV, §§ 392–399 (Props. I–III) · Book 5: Book IV: Areas of Polygons
57. Areas of rectangles
Goal: Your student can define area and prove that rectangles are as their bases and altitudes, and that the area of a rectangle is base times altitude.
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)
- Squared (graph) paper
What's in the book
Definitions of the unit of surface, area and equivalent figures (§§ 392–394), Propositions I–III (§§ 395–399), and Exercises 349–352.
The lesson, step by step
Warm up
Ask: What is the difference between equal figures and equivalent figures?
The new idea
The areas of two rectangles having equal altitudes are to each other as their bases.
The areas of two rectangles are to each other as the products of their bases by their altitudes.
The area of a rectangle is equal to the product of its base by its altitude.
Draw the figure
Draw a rectangle 7 cm by 4 cm on squared paper and count the unit squares (§ 399). Then draw two rectangles of the same height with bases 3 cm and 5 cm.
Walk through the proof
I: if the bases have a common measure, perpendiculars cut both rectangles into equal small rectangles, m in one and n in the other. If not, use limits as in Book II.
II: build a third rectangle S with the base of one and the altitude of the other, and multiply two ratios.
III: compare the rectangle with the unit square using Prop. II.
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Ex. 351. A square and a rectangle have the same perimeter, 100 yards. The length of the rectangle is 4 times its breadth. Compare their areas.
Ex. 352. On a certain map the linear scale is 1 inch to 5 miles. How many acres are represented on this map by a square the perimeter of which is 1 inch ?
For Ex. 352, one square mile is 640 acres.
Wrap up
Ask: A rectangle is 3 by 12 and a square has the same area. What is the side of the square? (6.)
Answers to the book's problems
Ex. 349: 18,144 to 1. Ex. 351: the square is 625 sq yd and the rectangle 400 sq yd, a ratio of 25 : 16. Ex. 352: 1000 acres.
Worked out for this site from the scan. If a number in your copy differs, trust the book.