## Cartesian Equation of a Circle

Since a circle is made up of all the points a fixed distance (its radius) from a given point (its centre) then the equation of a circle simply needs to ensure this is true. This can be done using Pythagoras’s theorem. This is because we can draw a right angled triangle with the centre of the circle at one corner and the point on the circle at the opposite corner as shown below. The radius is then the hypotenuse, the vertical side is the difference between the y co-ordinate of the point and that of the centre and the horizontal side is the difference between the x co-ordinate of the point and centre. From Pythagoras we therefore know that a circle of radius r and centre (a,b) must have a Cartesian equation

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

However, we can expand these brackets out to get

r^{2} = x^{2} – 2ax + a^{2} + y^{2} – 2by + b^{2}

but since a^{2}+b^{2}+r^{2}, -a and -b are all constant we can let

c = a^{2}+b^{2}-r^{2},

g = -a

f = -b

to get

**x ^{2} + y^{2} + 2gx + 2fy + c = 0**

where the circle has a centre (-g,-f) and radius √(a^{2}+b^{2}-c^{2})

## Area And Circumference of a Circle : pi

**This site is now at www.breakingwave.co.nr**

This is a basic guide to using pi to find the area and circumference of a circle using pi. And also explores why pi makes our formulae work.

area =πr^{2}

circumference = 2πr or πd

where r = radius and d=diameter

**Area**

^{2}. This is simple enough to use, we multiply the radius by itself and then by pi.

**Does this make sense?**

^{2}. Of course the circle’s area is a bit smaller so we need to find the ratio between the areas of the square and circle. If we then times this value by four we have a magic constant to multiply r squared by to find the area of a circle (we times by four because we need the area of 4 quadrants and r squared gives us one).

**Circumference**

**Does this also make sense?**

**Conclusion**

perimeter of square to circumference of circle.

By David Woodford

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