Circle

Definition: A circle is a simple shape, consisting of those points in a plane that are a given distance from a given point - the centre.

Origin: the center of a circle
the center of a circle

Radius: the distance from the center of a circle to any point on it.
Radius of a circle

Diameter: the longest distance from one end of a circle to the other. The diameter = 2 × radius.
Diameter

Circumference: the distance around the circle. It equals $\pi \times$ diameter.
Circumference

$\pi$ - pi: a number equal to 3.141592... or $\approx \frac{22}{7}$, that is $\frac{\text{the circumference}}{\text{the diameter}}$ of any circle.
pi-an animated example

Arc: a curved line that is part of the circumference of a circle.
an arc of a circle
The arc of the circle is measured in degrees or radians - for example: 90° or $\frac{\pi}{2}$ - a quarter of the circle, 180° or $\pi$ - a half of the circle.
The arc is smaller than 360°(or $2\pi$) because that is the whole circle.

Chord: a line segment within a circle that touches 2 points on the circle.
a chord of a circle

Sector: is like a slice of pie (a circle wedge).
Sector of a circle

Tangent of a circle: a line perpendicular to the radius that touches ONLY one point on the circle.
Tangent of a circle

Formulas

The formula for finding the circumference of a circle is $\pi \cdot \text{diameter} = 2 \cdot \pi \cdot \text{radius}$

The formula for finding the area of a circle is $\pi \cdot \text{radius}^2$

The standard notation for a radius is r, for a diameter - d, for a circumference - P and for area A.

$P = \pi \cdot d = 2 \cdot \pi \cdot r$
$A = \pi \cdot r^2$

Area of sector

circle sector
$r$ is the radius of the circle.
$\theta$ is the measurement of the central angle.
If the angle θ is in degrees, then the area = $\frac{\theta}{360} \pi r^2$
If the angle θ is in radians, then the area = $\frac{\theta}{2} r^2$

Angles

Central angle

Central angle

If the length of an arc is $\theta$ degrees or radians then the measurement of the central angle is also $\theta$(degrees or radians).

If you know the length of an arc(in inches, yards, foots, centimeters, meters...) you can find the measurement of its corresponding central angle($\theta$) by the formula:

$\theta = 360 \cdot \frac{l}{P} = \frac{360 \cdot l}{2 \cdot \pi \cdot r} = \frac{180 \cdot l}{\pi \cdot r}$

$l$ is the length of the arc.

Inscribed angle

An inscribed angle is formed by two chords and vertex on the circle.
$\angle APB$ is an inscribed angle.

inscribed angle
The measurement of the inscribed angle is equal to one-half the measurement of its intersected arc.

Example:
The arc $\widehat{AB}$ measures 84°. The measurement of $\angle APB = \frac{84}{2} = 42^\circ$

Angles between two secants

Case 1: two secants intersect inside a circle.

two hords
When two secants intersect inside a circle, the measurement of each angle formed is half the sum of the arcs.
In the figure, arc AB is 60° and the arc CD is 50°.
So angle 1 and 2 measure ½(60° + 50°) = 55°

Case 2: two secants intersect outside a circle.
secants intersecting outside a circle

The measurement of the angle formed is equal to half the difference of the arcs.

$\angle ABC =\frac{1}{2}(x - y)$

For example: if the bigger arc measures 80° and the smaller one 30° then $\angle ABC = \frac{1}{2}(80 - 30) = \frac{1}{2} \cdot 50 = 25^\circ$

Formula for intersecting chords

two chord intersecting
When two chords intersect inside a circle then:

$AX \cdot XB = CX \cdot XD$


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