The Area of a Circle

The Area of a Circle

 The area of a circle is a fundamental concept in geometry and is given by the formula A = πr^2, where A is the area of the circle and r is the radius of the circle. The value of π is approximately 3.14159 and is a mathematical constant that is defined as the ratio of the circumference of a circle to its diameter. In this proof, we will provide a simple and intuitive proof of the formula for the area of a circle that does not rely on calculus or other advanced mathematical concepts.

Consider a circle with radius r and circumference C. We can approximate the circle by inscribing it inside a regular polygon with n sides, where n is a large number. The inscribed polygon has side length s and perimeter P, which is equal to the circumference of the circle. The area of the polygon can be found by dividing it into n congruent triangles with base s and height h, where h is the distance from the center of the polygon to a side of the polygon.

By the Pythagorean theorem, we have h^2 + (s/2)^2 = r^2, since h is the hypotenuse of a right triangle with legs r and s/2. Solving for h, we get h = sqrt(r^2 - (s/2)^2). The area of each triangle is given by (1/2)sh, so the area of the polygon is given by:

A ≈ (1/2)nh(sqrt(r^2 - (s/2)^2))

As n gets larger, the polygon becomes more closely approximated to the circle, and the value of s approaches zero. We can use the limit as n approaches infinity to find the area of the circle. The perimeter of the inscribed polygon approaches the circumference of the circle, which is given by C = 2πr, and the value of s approaches zero, so we have:

A = lim (n → ∞) (1/2)nh(sqrt(r^2 - (s/2)^2))

= (1/2) lim (n → ∞) nh(sqrt(r^2 - (s/2)^2))

= (1/2) lim (n → ∞) nh(sqrt(r^2 - (r/n)^2))

= (1/2) lim (n → ∞) n(sqrt(r^2 - (r/n)^2))

We can simplify this expression by using the fact that the limit of a product is equal to the product of the limits, and the limit of a sum is equal to the sum of the limits. Thus, we have:

A = (1/2) lim (n → ∞) n(sqrt(r^2 - (r/n)^2))

= (1/2) r^2 lim (n → ∞) n(sqrt(1 - (1/n)^2))

= (1/2) r^2 lim (n → ∞) n(sinθ)

= (1/2) r^2 π

where θ is the angle formed by two adjacent sides of the inscribed polygon, which approaches zero as n gets larger, and we have used the fact that lim (n → ∞) n(sinθ) = π. Therefore, we have shown that the area of a circle is given by A = πr^2, which is a simple and intuitive result that does not rely on advanced mathematical concepts.