Ray's New Higher Arithmetic · Arts. 405–409 · Unit 28: Mensuration
121. Regular polygons and the circle
Goal: Your student finds the area of a regular polygon and the circumference and area of a circle, and sees where π comes from.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- A calculator, for checking only (optional)
- A string, a jar lid and a ruler
What's in the book
Regular polygons and the circle: area of a regular polygon, parts of a circle (chord, arc, segment, tangent, sector), how polygons of 6, 12, 24 sides close in on the circle, the value of π, general rules for the circle, sectors and segments, and examples (answers printed).
The lesson, step by step
Warm-up
Wrap a string around a jar lid, then measure across the lid. Divide. Do you get a bit more than 3?
Regular polygons
Read:
Any regular polygon may be divided into equal isosceles triangles
and the rule for the area:
Multiply the perimeter by half the apothem.
Parts of the circle
Read:
A tangent to a circle is a straight line having only one point in common with the curve; it simply touches the circle; a secant enters the figure from without.
The space inclosed by two radii and an arc, is called a sector
Where π comes from
The book starts with a hexagon inside a circle of radius 1 (perimeter 6), then doubles the sides to 12, 24, and on to 3072. The perimeter creeps up to 6.28318... Read:
The circle is regarded as composed of an infinite number of triangles whose common altitude is the radius and the sum of whose bases is the circumference.
Area of circle = ½ circumference × radius.
Pi
Read:
This important ratio, of circumference to diameter, is represented by the Greek letter π
The book usually uses 3.1416.
Guided practice
What are the circumferences whose diameters are 16, 22¼, 72.16, and 452 yd.?
Then find the area of a circle 10 ft. across.
Independent practice and wrap-up
Your student works more examples, including Example 4 (areas from circumferences, answers below) and the log puzzle. Ask: Which holds more, a square or a circle with the same 20-foot fence? (The circle.)
Afterwards
Two sittings: polygons, the circle and π; then the practice examples.
Answers to the book's problems
Ex. 4 (areas of circles from their circumferences; area = circumference² ÷ (4 × 3.1416)): 46 ft. → about 168.39 sq. ft.; 7 ft. 3 in. → about 4.18 sq. ft.; 6 yd. 1 ft. 4 in. (19⅓ ft.) → about 29.74 sq. ft.
Worked out for this site from the scan. If a number in your copy differs, trust the book.