In today's blog, I will go over Archimede's proof that the area of a circle is (1/2)rc. When this proof is combined with Euclid's proof (most likely from Eudoxus), it is possible to show that for all circles, the ratio of C/D is constant.
Postulate 1: For a given chord of a circle, the segment of the circumference is longer than the chord.
In the example above, BDC is longer than BC.
Postulate 2: For a given segment of the circumference, lines connected above it combined are longer.
In the example above, AG + GE is greater than the segment of the circumference AE.
Lemma 1: The area of a regular polygon is (1/2)h*Q where Q is the sum of the perimeter.
(1) A regular polygon can be divided up into a sum of triangles.
(2) The area of each triangle is equal to (1/2)*(GH)*(DE) [See here for details if needed]
(3) So, the area of the polygon = (# of sides)*(1/2)*(GH)*(DE)
(4) Let Q = the sum of the bases (for example, for the hexagon above, Q = DE + EF + FA + AB + BC + CD)
(5) Then, the area of the polygon = (1/2)*(GH)*Q
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. [See Lemma 2 and the Method of Exhaustion here for details]
(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.
Corollary 1: For all circles, the ratio of Circumference to Diameter is constant
(1) We know from Euclid (see here) that for any two circles C1 and C2 that:
Area1/Area2 = (Diameter1)2/(Diameter2)2
(2) We know from Theorem 1 above that:
Area of circle = (1/2)*radius*circumference.
(3) Combining step #1 with step #2 and using R=(1/2)D gives us:
[(1/2)*(1/2)D1*C1]/[(1/2)*(1/2)D2*C2] = [D1]2/[D2]2
(4) Canceling out (1/4)/(1/4) gives us:
[D1*C1]/[D2*C2] = [D1]2/[D2]2
(5) Multiplying both sides by (D2/D1) gives us:
C1/C2 = D1/D2
which means that:
C1*D2 = C2*D1
and finally that:
C1/D1 = C2/D2
Definition 1: π
From the Corollary, we know that the ratio of the circumference to the diameter is constant. π is this ratio from circumference to diameter. In other words, π = C/D.
Corollary 2: The area of a circle is πr2.
(1) From the theorem, the area of a circle is (1/2)(circumference)(radius)
(2) From the definition above, π = C/D
This means that C = D*π = 2*r*π
(3) Putting this all together gives us:
area = (1/2)(circumference)(radius) = (1/2)(2*r*π)(r) = πr2