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Area of a circle · 3 marks

Why were human beings so fond of using circular shapes? Was this only for practical reasons, or could there have been other reasons too? What kinds of uses have human beings found for the circular shape?

Answer: Practical: a circle is easy to draw with a rope and a peg, it rolls (wheels), it has no weak corners (pots, wells, domes), and it encloses the most area for a given boundary, saving wall or fence material. Other reasons: its perfect symmetry, the round sun and moon, beauty and religious or cultural meaning. Uses: wheels, pots, wells, round huts and granaries, coins, plates, bangles, clocks, gears, stupas and stadiums.

Step-by-step solution

Idea: A good answer gives practical reasons, at least one non-practical reason, and a list of uses, and if possible backs one reason with a calculation.

  1. Practical reasons. (1) Easy to make: tie a rope to a peg and walk round. (2) A wheel or log rolls smoothly because every point of the rim is the same distance from the centre. (3) No corners, so pressure is spread evenly: clay pots, wells and domes are strong. (4) For a fixed length of wall, a circle encloses the largest area.1 mark
  2. A check of (4): 88 m of fence as a square (side 22 m) encloses 22 × 22 = 484 m2; as a circle (radius 14 m, since 2 × 227 × 14 = 88) it encloses 227 × 14 × 14 = 616 m2. Same fence, about 27% more space.1 mark
  3. Other reasons and uses. The circle is perfectly symmetric and looks complete; people saw it in the sun, the moon and ripples, and used it in art, mandalas and sacred buildings. Uses: wheels and carts, pottery, wells, round huts, granaries and cylindrical towers, coins, plates, bangles, clocks, gears and pulleys, round tables (everyone equally close to the centre), arenas and stadiums.1 mark
People used circles for practical reasons (easy to draw, rolls, strong, most area for a given boundary) and for other reasons (symmetry, beauty, the sun and moon, cultural meaning). Uses include wheels, pots, wells, huts, granaries, coins, clocks and gears.

Answer to write in the exam

Practical: easy to draw (rope and peg); rolls; no corners (strong); largest area for a given perimeter.

Check: 88 m fence: square 22 × 22 = 484 m2; circle r = 14 m: 227 × 142 = 616 m2.

Other reasons: symmetry, beauty, sun and moon, cultural and religious meaning.

Uses: wheels, pots, wells, huts, granaries, coins, clocks, gears.

Common mistakes that cost marks

  • Giving only one reason. The question asks for practical reasons, other reasons and uses.
  • Claiming a circle has the largest perimeter for a given area. It is the other way: the smallest perimeter for a given area (largest area for a given perimeter).

How this can come in the exam

MCQ (1 mark)

A gardener has 88 m of fencing. Which shape of enclosure gives the greatest area?

  1. A square
  2. A circle
  3. A 30 m × 14 m rectangle
  4. An equilateral triangle
Show answer

(B) A circle
Square: 484 m2; rectangle: 420 m2; circle (r = 14 m): 616 m2; triangle: about 373 m2. The circle wins.

Try one yourself

A rope 44 m long is laid out first as a square and then as a circle. Compare the areas enclosed. (Use π = 227.)

Show answer

Square: 11 × 11 = 121 m2. Circle: r = 7, area 154 m2. The circle encloses 33 m2 more.

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