derivation of the equation of a circle?

Answer 1

If you are referring to an area of a circle then Archimedes did that a while ago by method of exhoustion. http://people.bath.ac.uk/slt20/archimedesmethod.html

However I think you are looking for a derivation in Cartesian coordinates.
1.Determine what is the relationship of a radius to the x and y.
From Pythagorean theorem x^2 + y^2 = r^2, where radius r is the hypotenuse.
2.The equation above is good but not general enough since it does not incorporates shifts in x and y axis. The x-a shifts to the right and x + a shifts the circle to the left. Similarly for y. then we have a general equation

(x-a)^2 + (y-b)^2=r^2


I hope that helps.

Answer 2

Are you asking for the derivative(slope) of the circle) or the equation of the circle

x^2 +y^2 = r^2
y^2 =r^2 -x^2

y =(r^2 -x^2)^(1/2) This will graph the top half of the circle. With the center of the circle being at (0,0) and the radius = r

The derivative is
1/2*(r^2-x^2)^(-1/2)*-2*x

y= -x*(r^2 -x^2)^(-1/2) This will give the slope of the circle at x

Answer 3

From the definition, the equation of a circle is
x² + y² = r²

If you mean differentiation, you can use implicit differentiation with r² as a constant
2x + 2y(dy/dx) = 0
2y(dy/dx) = -2x
dy/dx = -x/y

^_^

Answer 4

By the definition of circle, and Pythagoras' theorem.