Area of a Circle പരപ്പളവ്
We trapped the circumference between polygons. The same trick finds the area: polygons inside a circle fill it more and more completely as their sides increase, and the area of each polygon is easy to compute. The result is short and famous: area = ½ × circumference × radius = πr².
- Area of a triangle = ½ × base × height.
- Circumference c = 2πr (page 7 of this guide).
- Regular polygons in a circle come closer and closer to the circle as the number of sides grows (page 5).
1. Polygons fill the circle
The yellow part is the circle's area not yet covered by the blue polygon. As n grows, it shrinks towards nothing, so the polygon areas come closer and closer to the circle's area.
2. The area of a regular polygon: ½ × perimeter × h
3. From polygons to the circle (page 160)
വൃത്തത്തിന്റെ പരപ്പളവ്, അതിന്റെ ചുറ്റളവിന്റെയും ആരത്തിന്റെയും ഗുണനഫലത്തിന്റെ പകുതിയാണ്
The area of a circle is half the product of its circumference and its radius: a = ½cr.
Since c = 2πr: a = ½ × 2πr × r = πr².
വൃത്തത്തിന്റെ പരപ്പളവ്, ആരവർഗത്തിന്റെ π മടങ്ങാണ് – the area of a circle is π times the square of its radius. Example: radius 5 cm ⇒ area 25π sq cm ≈ 78.54 sq cm.
4. Another way to see it: slices into a rectangle
another way to see it Cut the circle into equal slices (sectors) and lay them alternately point-up and point-down. The more slices, the more the shape looks like a rectangle whose length is half the circumference and whose height is the radius.
Length ≈ half the circumference = πr, height ≈ r, so area ≈ πr × r = πr² – the same answer.
5. Rings വൃത്തവലയം
Ring area = πR² − πr² = π(R² − r²). In a right triangle with hypotenuse R and one side r, the third side t has t² = R² − r², so the circle of radius t has exactly the ring's area.
Draw a circle of radius √(10/π) with Circle: Centre & Radius (type sqrt(10/pi)) and measure it with the Area tool: the area is 10. Then try to draw a circle of area 25 (radius √(25/π) ≈ 2.82).
A 12-inch pizza (radius 6) has area 36π; an 8-inch pizza (radius 4) has 16π. One 12-inch pizza is 2¼ times one 8-inch pizza, and more than two 8-inch pizzas together (32π). Area grows with the square of the radius.
6. Check yourself
- A regular polygon in a circle has area ½ × perimeter × h (h = perpendicular from the centre to a side).
- As the sides increase, perimeters approach c and h approaches r, so the polygon areas approach ½cr.
- Area of a circle = ½cr = πr². Radius 5 ⇒ 25π.
- Ring between radii R and r: π(R² − r²); it equals a circle whose radius is the third side of a right triangle with hypotenuse R and side r.