why the area of a circle is pi r square? how the value of pi viz 22/7 is derived?

Answer 1

Pi, 22/7, can be calculated as the ratio of circumference of a circle to the diameter. What this means is circumference is pi times the diameter.

Since the circumference happens to be pi*dia (or 2*pi*r), integrating this between 0 and r will give the area of the circle. This will be pi r square.

Answer 2

How's your calculus? If you do an integral of a circle in radial coordinates, it works out that way. It's not much of an answer, though...

Pi is defined as the ratio of a circles circumference to its diameter. 22/7 is an old approximation that is still used because it's easy to remember. In actuality, it's infinitely long and non-repeating. The actual value can be calculated in a variety of ways. Here's a sample: 3.14159265358979323846264338327950288419716939937510

Answer 3

let be the circle C(r, O). to approximate the area of this circle you divide it into n isosceles rectangles A(n)OB(n). the height of this rectangle is [OH(n)] = h(n), [OA(n)] = [OB(n)] = r. the area of one rectangle is (h * [A(n)B(n)])/2. The area of the circle is approximated by (n * h * [A(n)B(n)])/2.

lim (n * h(n) * [A(n)B(n)])/2 = pi* r ^ 2

how:

lim(n * [A(n)(B(n)]) = l (the length of the circle) because when we divide the circle into an infinite number of triangles the sum of the lengths of the bottom segments equals the circumference of the circle witch is 2*pi*r.

lim h(n) = r because the height of these triangles becomes equal to r.

therefor the area of the circle = (2 * pi * r * r ) / 2 = pi * r^2

pi = l / 2 * r witch is constant so the area of the circle is (l * r ^2) / 2 * r = l * r / 2 (kinda like a triangle)

22/7 is merely an approximation just for easy work

Answer 4

The value 22/7 come out of the beginning of the continued fraction of pi. It can be easily derived by simply noting that
x_1 = 1/(pi-3) = 7.0625...
so pi is near to
3+1/7 = 22/7
If you go on you will find a very good approximation for pi:
x_2 = 1/((x_1)-7) = 15.9966 very near to 16, so
pi is near to
3+1/(7+1/16) = 355/113

and so on for even better approximations.

Answer 5

A circle is a form of ellipse, denoted by the equation

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.

Solving the equation of the ellipse for y, the following is derived:

\frac{y^2}{b^2} = 1 - \frac{x^2}{a^2} = \frac{a^2 - x^2}{a^2} \ or \ y = \pm\frac{b}{a}\sqrt{a^2 - x^2}

Because the ellipse is symmetric with respect to both axes, the total area A is four times the area in the first quadrant. The part of the ellipse in the first quadrant is given by the function
y = \frac{b}{a}\sqrt{a^2 - x^2} \ 0 \le x \le a and so:\ \frac{1}{4} A = \int_0^a \frac{b}{a} \sqrt{a^2 - x^2} \,dx

To evaluate this integral we substitute x = asinθ. Then dx = acosθdθ. To change the limits of integration we note that when x = 0, sinθ = 0, so θ = 0; when x = a, sinθ = 1, so \theta = \frac{\pi}{2}.

Also

\sqrt{a^2 - x^2} \ = \ \sqrt{a^2 - a^2 \sin^2 \theta} \ = \ \sqrt{a^2 \cos^2 \theta} \ = \ a |\cos \theta| \ = \ a \cos \theta

since 0 \le \theta \le \frac{\pi}{2}

Therefore

A = 4 \frac{b}{a} \int_0^a \sqrt{a^2 - x^2} \,dx \ = 4 \frac{b}{a} \int_{0}^{\pi/2} a \cos \theta \cdot a \cos \theta \,d\theta

= 4ab \int_{0}^{\pi/2} \cos^2 \theta \,d\theta \ = 4ab \int_{0}^{\pi/2} \frac{1}{2} (1 + \cos 2\theta)

= 2ab[\theta + \frac{1}{2} \sin 2\theta]_{0}^{\pi/2} \ = 2ab (\frac{\pi}{2} + 0 - 0)

\ \ \ = \pi ab

This shows that the area of an ellipse with semiaxes a and b is πab. In particular, taking a = b = r, the famous formula for the area of a circle with radius r, A = πr2, has been proven.
The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = 25/8, which is within 0.5% of the true value.

The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.

It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular.
Principle of Archimedes' method to approximate π.
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Principle of Archimedes' method to approximate π.

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 223/71 and 22/7. The average of these two values is roughly 3.1419.

The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.

The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.

The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355/113 and 22/7, in the 5th century.

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of π /4 into the form

π = √12 (1 - 1/(33) + 1/(532) - 1/(733) + ...

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π /4, he was able to compute π to an accuracy of 13 decimal places.

The astronomer Ghyath ad-din Jamshid Kashani (1350-1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2π = 6.2831853071795865

The German mathematician Ludolph van Ceulen in 1615 computed the first 32 decimal places of π. He was so proud of this accomplishment that he had them inscribed on his tombstone.

In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).

Answer 6

pi, which is the ratio of the circumference to the diameter of a circle, was found to be a constant for all circles.

With this constant, pi, the area of the circle was derived.

Answer 7

22/7 is a very bad approximation that is only accurate to two decimal places. Just remember 3.14159, and you can calculate anything you need to a high degree of accuracy.

Answer 8

Pie are NOT square. Pie are round. Cornbread are square.....

Answer 9

i cant do better than that ^^^^^^^^^^^^^^^^^^