Hii , this is the thread for simple derivations on geometrical figures.
Here I begin with the easiest proof of the area of circle. Although there are many proofs but easiest is that , which I am telling you all ( I think ) .
1. The proof by Greeks 2000 years ago ! : http://www.slideshare.net/yaherglani...-proof-1789707
2. The analytical proof :
Divide the circumference into n equal parts each of length l
=> nl = 2π r => l = 2π r/n
When n is very large and l very small, each sector of the circle formed by the small arc of length l is like a triangle and
its area = (1/2) base x altitude
= (1/2) l * r
= (1/2) 2π r/n * r
and area of the circle
= n * area of each sector
= n * [ (1/2) 2π r/n * r ]
= π r^2.
3. The proof which I discovered !
View image: proof
4. Proof of Archimedes by Dr. Math : Math Forum - Ask Dr. Math
5. Euclid's Proof by Dr. Math : Math Forum - Ask Dr. Math
6. Archimedes' simplified proof :
Theorem: The area of a circle is (1/2)circumference * radius
(1) Let K = (1/2)*C*R where C = circumference and R = radius.
(2) Let A = the area of the circle.
(3) Assume that A is greater than K
(4) We can inscribe a polygon inside A that is greater than K and less than A.
(5) So, the area of the polygon = (1/2)*Q*h where h is the distance from the center to the base and where Q is the perimeter of the polygon. [See Lemma 1 above]
(6) But Q is less than C (see Postulate 1 above) and h is less than R.
(7) So we have Area Polygon = (1/2)Q*h which is less than (1/2)*C*R
(8) But this contradicts step #4 so we reject step #3.
(9) Now, let's assume that K is greater than A.
(10) We can circumscribe a polygon P around A such that P is greater than A but less than K. [See Lemma 3 and Method of Exhaustion here for details.]
(11) From Lemma 1 again, we know that the area of this polygon is (1/2)*Q*h where Q is the perimeter of the polygon and h is the height.
(12) In the case of the circumscribed polygon (see diagram for Postulate 2), h = R.
(13) Using Postulate 2 above, we see that Q is greater than C.
(14) But then the area of the polygon is greater than K since (1/2)*Q*R is greater than (1/2)*C*R
(15) But this contradicts step #10 so we reject our assumption at step #9.
(16) Now, we apply the Law of Trichomoty (see here) and we are done.
(17) From the theorem, the area of a circle is (1/2)(circumference)(radius)
(18) From the definition above, π = C/D
This means that C = D*π = 2*r*π
(19) Putting this all together gives us:
area = (1/2)(circumference)(radius) = (1/2)(2*r*π)(r) = πr2
1. Proof of the area of a circle by inscribing a polygon :
2. By inscribing a regular polygon
Evaluating pi. The area of a circle. Topics in trigonometry
3. Another proof :
4. By using equation of circle :
5. By calculus :
6. Other proofs :
Please share more proofs on geometrical figures , 3rd proof I discovered myself though .
Regards - Sankalp