Summary & Quick Quiz പുനരവലോകനം
Every formula of the chapter in one place, the idea behind them, the story of π, the mistakes to avoid, flashcards, a scored quiz and your study checklist.
1. The chapter in three pictures
2. All the formulas
Radius (ആരം) is centre to edge; diameter (വ്യാസം) is edge to edge through the centre.
വൃത്തങ്ങളുടെ ചുറ്റളവുകൾ മാറുന്നത്, വ്യാസങ്ങൾ മാറുന്ന അതേ തോതിലാണ്
Circumferences change in the same proportion as diameters.
വൃത്തത്തിന്റെ ചുറ്റളവ്, അതിന്റെ വ്യാസത്തിന്റെ π മടങ്ങാണ്
Radius 5 m ⇒ 10π ≈ 31.416 m.
Circumference of the circle of diameter 1. Not a fraction, not even made of roots. 22/7 ≈ 3.1429 is only close.
Regular polygon in a circle: perimeter p, perpendicular h from the centre to a side.
വൃത്തത്തിന്റെ പരപ്പളവ്, അതിന്റെ ചുറ്റളവിന്റെയും ആരത്തിന്റെയും ഗുണനഫലത്തിന്റെ പകുതിയാണ്
Radius 5 cm ⇒ 25π sq cm.
Radii 2.5 and 2: 6.25π − 4π = 2.25π, the same as a circle of radius 1.5 (Pythagoras).
Twice the radius: twice the circumference, four times the area.
We could not measure a circle directly, so we drew regular polygons inside it with more and more sides. Their perimeters scale with the diameter and close in on the circumference – so circumference ∝ diameter, and the constant is π. Their areas are ½ph, and as p → c and h → r they close in on ½cr – so the circle's area is ½cr = πr².
- Egypt (Ahmes papyrus, ~1500 BC): 256/81 ≈ 3.16. Babylon: 25/8 = 3.125. India (Shatapatha Brahmana, ~7th century BC): 339/108 ≈ 3.139.
- Archimedes (3rd century BC), with 96-sided polygons: 3.1408 < π < 3.1428.
- Zu Chongzhi (China, 5th century CE), with 12 288-sided polygons: 3.1415926 < π < 3.1415927.
- Madhavan of Sangamagrama (Kerala, 14th century): 1 − 1/3 + 1/5 − 1/7 + … → π/4, and π ≈ 3.14159265359. The series was rediscovered by Gregory and Leibniz in the 17th century.
- π is not a fraction – proved in the 18th century. Computers now know trillions of its digits.
3. Common mistakes
- Using the diameter in πr². A circle of diameter 10 cm has area π × 5² = 25π, not 100π.
- Mixing units. Circumference is a length (cm, m); area is in square units (sq cm, sq m).
- 22/7 is not exactly π. It is a handy approximation (≈ 3.1429). Use it only when the question says so; otherwise leave the answer in terms of π.
- "Double the radius, double the area" is wrong. The area becomes four times; only the circumference doubles.
- Ring area is not π(R − r)². It is πR² − πr² = π(R² − r²).
- Rounding too early. Keep π in the answer until the last step, then round once.
4. Flashcards
5. Scored quiz
6. Your study checklist
- Circumferences change in the same proportion as diameters; the constant is π ≈ 3.1416.
- c = πd = 2πr.
- a = ½cr = πr²; a ring is π(R² − r²).
- Both formulas come from regular polygons with more and more sides closing in on the circle.
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