# Circle **The circle is a set of points that are equidistant from a single point, the center.** It can be thought of as a **special kind of [[Ellipse]], one whose two focal points are the same**. It can also be defined as a [[Conic Section]] produced when intersecting a cone with a plane that is perpendicular to the cone's center line. See [[Trigonometric Functions]] for functions that define relationships between a circle's $x$ and $y$ coordinates. A circle can be defined algebraically as a set of points that satisfy the following equation: $ (x-n)^2 + (y-m)^2 = r^2 $ Where point $(n,m)$ is the center point and $r$ the radius of the circle. ## Area of a Circle $A = r^2\pi$ You can reason about this by think of a circle as being divided into slices like such: <div> <img src="https://imgur.com/jVYHYUd.png" style="background: white; padding: 10px; width: 200px" /> <img src="https://imgur.com/xwEJqlO.png" style="background: white; padding: 10px; width: 300px; top: 50%; transform: translateY(-50%)" /> </div> If we let the number of slices tend to $\infty$ the rearranged slices will form a proper parallelogram (the right image), whose area will be $A = bh$, where $b$ is the parallelogram's base, in this case half the circle's circumference, and $h$ is the parallelogram's height, in this case the circle's radius. From there: $A = bh = \frac 1 2 2r\pi \cdot r = r^2\pi$ Another way of thinking about the area is as a sum of areas of concentric rings. If you straighten a ring you get a trapezoid (because the inner radius is smaller), with the height $h$, where $h$ represents the thickness of the ring. As the number of rings approaches infinity, the trapezoids approximate rectangles which can be arranged from the smallest one to the largest one. As the number of rings increases, the shapes fit closer and closer to that of a triangle whose height corresponds to the length of the largest ring ($2r\pi$), and whose base is $r$ (thickness of each of the rings times the number of rings). From there (area of a triangle) we get: $A = \frac 1 2 2r \pi r$. This is a great example of how calculus can be used to solve similar geometric questions: ![[The Essence of Calculus#^2fl1p7]] ![[The Essence of Calculus#^t6xmvd]] ![[The Essence of Calculus#^fjfdnj]] As per above, as the circumference of a circle is $2\pi r$, we can say that: $A(r) = \int_0^r 2\pi r \, dr = \pi r^2$ ## Area of a Sector A sector is defined as a portion of a circle (a slice). $A = \frac 1 2 \theta r^2$ You can reason with this by extending the reasoning used for the area of the full circle. In case of a sector, the rearranged parallelogram will have a base of half $\theta$ radians, with its height being the radius of the sector. The calculus reasoning for the area of the circle above makes it even easier to see that the area of a sector is simply $A_s(r) = \int_0^r \theta r \,dr = \frac 1 2\theta r^2$