(See the second graph on the right for diagram) The circle is the set of points on a plane that are equidistant from a given point $O$. The distance from the center to the set of the points is $r$ which is also called the radius. The point $O$ is called the center. Twice the radius is called the diameter $d=2r$. The angle a circle makes from its center is a full circle, which is equal to $360^{\circ}$ or $2 \pi$ radians. The study of the circle goes back beyond recorded history. The invention of the wheel is a fundamental discovery of properties of a circle. The Greeks considered the Egyptians as the inventors of geometry. The scribe Ahmes, the author of the Rhind papyrus, gives a rule for determining the area of a circle which corresponds to $\pi=\frac{256}{81}$ which is approximately $3.16049382716....$. The first theorems relating to circles are attributed to Thales around 650 BC. Book III of Euclid's Elements deals with properties of circles and problems of inscribing and escribing polygons.One of the problems of Greek mathematics was the problem of finding a square with the same area as a given circle. Several of the 'famous curves' in this stack were first studied in an attempt to solve this problem. Anaxagoras in 450 BC is the first recored mathematician to study this problem. The problem of finding the area of a circle led to integration. For the circle with formula given above the area is $\pi a ^2$;and the length of the curve is $2 \pi a$. The pedal of a circle is a cardioid if the pedal point is taken on the circumference and is a limacon if the pedal point is not on the circumference. The caustic of a circle with radiant point on the circumference is a cardioid, while if the rays are parallel then the caustic is a nephroid. A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area. The perimeter $C$ of a circle is called the circumference, and it can be found by $C=\pi d=2\pi r$. Remember, this can be found by using calculus using the formula for arc length in polar coordinates. The formula would end up being: $C=\int_{0}^{2\pi}\sqrt{r^2+\left ( \frac{dr}{d\theta} \right )^2}d\theta$. However, $r\left ( \theta \right )=r$, the formula ends up becoming $C=\int_{0}^{2\pi}r\; d\theta=2\pi r$. The area of the circle can be also found by using Calculus. You can see how to do it in the video section on the right. The circle can be seen everywhere, from wheels to cups. **Click "Read More" to see the interactive graphic** | Equations*: Cartesian Equation: $\displaystyle x^2+y^2=a^2$ $\displaystyle y=\pm \sqrt{a^2-x^2}$ Polar Equation: $\displaystyle r=a$ Parametric Equations: $\displaystyle \left ( \begin{matrix} x\left ( t \right )\\ y\left ( t \right ) \end{matrix} \right )=\left ( \begin{matrix} a \cos t\\ a \sin t \end{matrix} \right )$ * Note: let the radius be $a$ Graphs: Videos: |

**Interactive Graphic:**

**References:**

- "Circle." Wikipedia. Wikimedia Foundation, 22 Feb. 2014. Web. 27 Feb. 2014.
- Weisstein, Eric W. "Circle." Wolfram MathWorld. Wolfram Research, Inc., 1 Dec. 2004. Web. 01 Mar. 2014.
- Wolfram|Alpha. "Circle." Wolfram|Alpha. Wolfram Research, Inc., 01 Mar. 2014. Web. 01 Mar. 2014.