A circle is a shape consisting of all continuous points in a Plane which are at a given distance from a given Point, the centre. This distance is fixed and is called the radius.
Nomenclature:
- Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
- Centre: the point equidistant from all points on the circle.
- Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
- Circumference: the length of one period along the circle, or the distance around the circle.
- Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
- Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted and required to be a positive number. A circle with is a degenerate case consisting of a single point.
- Secant: an extended chord, a coplanar line, intersecting a circle in two points.
- Tangent: a coplanar straight line that has one single point in common with a circle (“touches the circle at this point”).
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
% Define points
\coordinate (o) at (0,0); % Center
\coordinate (A) at (3,0); % Rightmost point (circle edge)
\coordinate (B) at (2.1,2.1); % Radius endpoint
\coordinate (C) at (-1.8,-2.4); % Chord start
\coordinate (D) at (2.4,-1.8); % Chord end
\coordinate (E) at (-3.5,2.2); % Secant start
\coordinate (F) at (3.5,-1.5); % Secant end
\coordinate (T1) at (3,-2); % Tangent start (orthogonal fix)
\coordinate (T2) at (3,2); % Tangent end (orthogonal fix)
% Draw the circle
\draw[thick] (o) circle(3);
\fill[black!20] (o) circle(0.1) node[below] {center};
% Draw radius
\draw[green!20, thick] (o) -- (B) node[midway, above, sloped] {Radius};
% Draw diameter
\draw[teal, thick] (-3,0) -- (3,0) node[near end, above] {Diameter};
% Draw chord
\draw[blue!20, thick] (C) -- (D) node[midway, above, sloped] {Chord};
% Draw secant
\draw[blue!40, thick] (E) -- (F) node[pos=0.4, below left, sloped] {Secant};
% Draw tangent (now perpendicular to the diameter)
\draw[purple!40, thick] (T1) -- (T2) node[midway, right] {Tangent};
% Mark the right angle
\draw[dashed] ($(A) - (0.3,0)$) -- ($(A) - (0.3, 0.3)$) -- ($(A) - (0,0.3)$);
\end{tikzpicture}
\end{document}Area:
Perimeter:
We can also define the sine and cosine [trig functions](Trigonometry#Definition of trigonometric functions) using the circle. Say we have a unit circle (radius = 1) i.e.
\begin{document}
\begin{tikzpicture}[scale=4]
% draw the coordinates
\draw[->] (-1.5cm,0cm) -- (1.5cm,0cm) node[right,fill=white] {$x$};
\draw[->] (0cm,-1.5cm) -- (0cm,1.5cm) node[above,fill=white] {$y$};
% draw the unit circle
\draw[thick] (0cm,0cm) circle(1cm);
\foreach \x in {0,30,...,360} {
% lines from center to point
\draw[gray] (0cm,0cm) -- (\x:1cm);
% dots at each point
\filldraw[black] (\x:1cm) circle(0.4pt);
% draw each angle in degrees
\draw (\x:0.6cm) node[fill=white] {$\x^\circ$};
}
% draw each angle in radians
\foreach \x/\xtext in {
30/\frac{\pi}{6},
45/\frac{\pi}{4},
60/\frac{\pi}{3},
90/\frac{\pi}{2},
120/\frac{2\pi}{3},
135/\frac{3\pi}{4},
150/\frac{5\pi}{6},
180/\pi,
210/\frac{7\pi}{6},
225/\frac{5\pi}{4},
240/\frac{4\pi}{3},
270/\frac{3\pi}{2},
300/\frac{5\pi}{3},
315/\frac{7\pi}{4},
330/\frac{11\pi}{6},
360/2\pi}
\draw (\x:0.85cm) node[fill=white] {$\xtext$};
\foreach \x/\xtext/\y in {
% the coordinates for the first quadrant
30/\frac{\sqrt{3}}{2}/\frac{1}{2},
45/\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
60/\frac{1}{2}/\frac{\sqrt{3}}{2},
% the coordinates for the second quadrant
150/-\frac{\sqrt{3}}{2}/\frac{1}{2},
135/-\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
120/-\frac{1}{2}/\frac{\sqrt{3}}{2},
% the coordinates for the third quadrant
210/-\frac{\sqrt{3}}{2}/-\frac{1}{2},
225/-\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
240/-\frac{1}{2}/-\frac{\sqrt{3}}{2},
% the coordinates for the fourth quadrant
330/\frac{\sqrt{3}}{2}/-\frac{1}{2},
315/\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
300/\frac{1}{2}/-\frac{\sqrt{3}}{2}}
\draw (\x:1.25cm) node[fill=white] {$\left(\xtext,\y\right)$};
% draw the horizontal and vertical coordinates
% the placement is better this way
\draw (-1.25cm,0cm) node[above=1pt] {$(-1,0)$}
(1.25cm,0cm) node[above=1pt] {$(1,0)$}
(0cm,-1.25cm) node[fill=white] {$(0,-1)$}
(0cm,1.25cm) node[fill=white] {$(0,1)$};
\end{tikzpicture}
\end{document}If we take a line from the origin and intersect it with the unit circle at some point, then take the angle made from this line (and the of the x-axis). We find we can use this to define the trig functions:
\begin{align} \sin(\theta) &= y \\ \cos(\theta) &= x \end{align} $$From this and our previous definition of the unit circle, it follows that:\sin^2(\theta) + \cos^2(\theta) = 1
which is the [Pythagorean identity](Trigonometry##Identities and Laws).