# Circle

Radius of a circle: $$R$$
Diameter of a circle: $$D$$
Chord length: $$a$$
Chord segments: $${a_1},$$ $${a_2},$$ $${b_1},$$ $${b_2}$$
Arc lengths: $$s,$$ $${s_1},$$ $${s_2}$$
Secant segments: $$e,$$ $${e_1},$$ $$f,$$ $${f_1}$$
Coordinates of the centre of a circle: $${x_0},$$ $${y_0}$$
Coordinates of a point of a circle: $$x,$$ $$y$$
Height of the segment of a circle: $$h$$
Central angle: $$\alpha$$, $$x$$
Inscribed angle: $$\beta$$
Angle between two chords: $$\varphi$$
Angle between two secants: $$\gamma$$
Angle between secant and tangent: $$\delta$$
Angle between tangent and chord: $$\theta$$
Angle between two tangents: $$\eta$$
Perimeter: $$P$$
Area: $$S$$
1. A circle is the set of all points in a plane equidistant from a given point called the center of the circle. The distance between the centre and any point of the circle is called the radius of the circle.
2. A line segment connecting two points of a circle is called the chord. A chord passing through the centre of a circle is a diameter. The diameter of a circle is twice as long as the radius:
$$D = 2R$$
3. A central angle is an angle whose vertex coincides with the centre of the circle. The relation between the chord length $$a$$ and the central angle $$\alpha$$ is given by
$$a = 2R\sin {\large\frac{\alpha }{2}\normalsize}$$
4. An arc of a circle is the portion of the circle between two given points. The measure of an arc (in degrees or radians) is the measure of the central angle subtended by this arc. The length of the arc is given by the formula
$$s = \alpha R$$,
where $$\alpha$$ is the central angle in radians, $$R$$ is the radius of the circle.
5. An inscribed angle of a circle is an angle whose vertex lies on the circle and the sides of the angle are chords of the circle. If an inscribed angle and a central angle of a circle subtend the same arc, the inscribed angle is half the central angle:
$$\beta = {\large\frac{\alpha }{2}\normalsize},$$
6. The point of intersection of two chords divides each chord into segments whose product is constant:
$${a_1}{a_2} = {b_1}{b_2}$$
7. The angle between two chords is equal to half the sum of the intercepted arcs:
$$\varphi = {\large\frac{{{s_1} + {s_2}}}{2}\normalsize},$$
where $${s_1}$$, $${s_2}$$ are the measures of the arcs (in degrees or radians).
8. A secant of a circle is a line drawn from a point outside the circle that intersects the circle at two points. For any two secants, the product of the external segment and the whole length of the first secant is equal to the product of the external segment and the length of the second secant:
9. The angle between two secants drawn from a point outside the circle is equal to half the difference of the enclosed arcs:
$$\gamma = {\large\frac{{{s_1} – {s_2}}}{2}\normalsize},$$
where $${s_1}$$, $${s_2}$$ are the measures of the corresponding arcs (in degrees or radians).
10. For any secant and tangent drawn from a point outside the circle, the product of the whole length of the secant and its external portion is equal to the square of the length of the tangent:
$$f{f_1} = {g^2}$$
11. The angle between a secant and a tangent drawn from a point outside the circle is equal to half the difference of the enclosed arcs:
$$\delta = {\large\frac{{{s_1} – {s_2}}}{2}\normalsize}$$,
where $${s_1}$$, $${s_2}$$ are the measures of the corresponding arcs.
12. The angle between a tangent and a chord drawn from the point of tangency is equal to half the measure of the arc subtended by the chord:
$$\theta = {\large\frac{s}{2}\normalsize} = {\large\frac{\alpha }{2}\normalsize}$$
13. A tangent to a circle is always perpendicular to the radius at the point of tangency.
14. The angle between two tangents drawn from a point is equal to half the difference of the enclosed arcs:
$$\eta = {\large\frac{{{s_1} – {s_2}}}{2}\normalsize}$$,
where $${s_1}$$, $${s_2}$$ are the measures of the corresponding arcs (in degrees or radians).
15. Equation of a circle in Cartesian coordinates
$${\left( {x – {x_0}} \right)^2}$$ $$+\;{\left( {y – {y_0}} \right)^2}$$ $$= {R^2},$$
where $${x_0}$$, $${y_0}$$ are the coordinates of the centre of the circle, $$R$$ is the radius, $${x,y}$$ are the coordinates of a point on the circle.
16. Perimeter of a circle
$$P = 2\pi R = \pi D$$
17. Area of a circle
(also called the area of a disk)
$$S = \pi {R^2} = {\large\frac{{\pi {D^2}}}{4}\normalsize}$$ $$= {\large\frac{{PR}}{2}\normalsize}$$
18. A circular sector is the portion of a circle enclosed by two radii and the corresponding arc.
19. Perimeter of a sector
$$P = s + 2R$$,
where $$s$$ is the length of the arc, $$R$$ is the radius of the circle.
20. Area of a sector
$$S = {\large\frac{{Rs}}{2}\normalsize} = {\large\frac{{{R^2}x}}{2}\normalsize}$$ $$= {\large\frac{{\pi {R^2}\alpha }}{{360^\circ}}\normalsize},$$
where $$s$$ is the arc length, $$R$$ is the radius of the circle, $$x$$ is the central angle in radians, $$\alpha$$ is the central angle in degrees.
21. A circular segment is the portion of a circle enclosed by bounded an arc and a chord joining the endpoints of the arc.
22. Height of a segment
$$h = R$$ $$-\; {\large\frac{1}{2}\normalsize}\sqrt {4{R^2} – {a^2}} ,$$ $$h \lt R$$
23. Relationship between the height of a segment and the chord length
$$a = 2\sqrt {2hR – {h^2}}$$
24. Perimeter of a segment
$$P = s + a$$,
where $$s$$ is the arc length, $$a$$ is the chord length.
25. Area of a segment
$$S = {\large\frac{1}{2}\normalsize}\left[ {sR – a\left( {R – h} \right)} \right] =$$ $${\large\frac{{{R^2}}}{2}\normalsize}\left( {{\large\frac{{\alpha \pi }}{{180^\circ}}\normalsize} – \sin \alpha } \right)$$ $$= {\large\frac{{{R^2}}}{2}\normalsize}\left( {x – \sin x} \right)$$,
where $$s$$ is the arc length, $$a$$ is the chord length, $$h$$ is the height of the segment, $$R$$ is the radius of the circle, $$x$$ is the central angle in radians, $$\alpha$$ is the central angle in degrees.
26. Approximate formula for the area of a segment
$$S \approx {\large\frac{{2ha}}{3}\normalsize}$$.
Here $$h$$ is the height of the segment, $$a$$ is the chord length.