The following are short descriptions of the circle shown below.

Tangent
Tangent is a line that would pass through one point on the circle.
Secant
Secant is a line that would pass through two points on the circle.
Chord
Chord is a secant that would terminate on the circle itself.
Diameter, d
Diameter is a chord that passes through the center of the circle.
Radius, r
Radius is one-half of the diameter.

 

The Circle

 

Area of the circle
$A = \pi \, r^2$

$A = \frac{1}{4}\pi \, d^2$

 

Circumference of the circle
$c = 2\pi \, r$

$c = \pi \, d$

 

Sector of a Circle
circular-sector.jpgLength of arc:

$s = \dfrac{\pi \, r \theta_{degree}}{180^\circ}$

$s = r \, \theta_{radians}$
 

Area of the sector:
$A = \dfrac{\pi \, r^2 \theta_{degrees}}{360^\circ}$

$A = \frac{1}{2}r^2 \, \theta_{radians} $

$A = \frac{1}{2}sr$
 

Segment of a Circle
Area of circular segment with s < ½ c:
$A = A_{sector} - A_{triangle}$

$A = \frac{1}{2}r^2 (\theta_{radian} - \sin \theta_{degrees})$
 

segment-of-a-circle.jpg

 

Area of circular segment with s > ½ c:
$A = A_{sector} + A_{triangle}$

$A = \frac{1}{2}r^2 (\alpha_{radian} + \sin \theta_{degrees})$

 

Relationship Between Central Angle and Inscribed Angle

Central angle
Angle subtended by an arc of the circle from the center of the circle.
Inscribed angle
Angle subtended by an arc of the circle from any point on the circumference of the circle. Also called circumferential angle and peripheral angle.

 

Figure below shows a central angle and inscribed angle intercepting the same arc AB. The relationship between the two is given by
 

$\alpha = 2\theta \, \text{ or } \, \theta = \frac{1}{2}\alpha$

 

if and only if both angles intercepted the same arc. In the figure below, θ and α intercepted the same arc AB.
 

Inscribed and Central Angles

 

Watch our video below for proof of this relationship.
 

Applications of the Relationship Between Central Angle and Inscribed Angle

Right Triangle Inscribed in a Circle
The hypotenuse of triangle inscribed in a circle coincides with the diameter of the circle.
 
Right triangle inscribed in a cirlce

 

We can also say that an angle inscribed in a semicircle is a right angle. From the figure above, the diameter AC is the hypotenuse of triangles AB1C, AB2C, AB3C, and AB4C.

Intersecting Chords
From the figure below, chords AC and BD intersect at E. Angle DAC and angle DBC intercepted the same arc CD, therefore, both angles are equal to one-half of the central angle DOC (not shown in the figure). We denote θ for angles DAC and DBC. Angle β = angle ACB = angle ADB, intercepting the arc AB. Triangle ADE is therefore similar to triangle BCE. By ratio and proportion of these similar triangles
 
Intersecting Chords

 

$\dfrac{opposite\,\,to\,\,\theta}{opposite\,\,to\,\,\beta} = \dfrac{DE}{AE} = \dfrac{CE}{BE}$

$BE \times DE = AE \times CE$
 

This means that for intersecting chords in a circle, the product of segments of one is equal to the product of segments of the other.

Intersecting Secants
Secant lines ED and EC intersect at point E as shown below. Angles ADB and ACB intercepted the same arc AB, therefore the angles are equal and we denote both by β. Also, angles DAC and DBC intercepted a common arc CD, both angles are equal and denoted as θ. Finally, angles EAC and EBD are both 180° - θ and denoted as Ø.
 
Intersecting Secants

 

Therefore, triangles EAC and EBD are similar, and by ratio and proportion of similar triangles

$\dfrac{opposite\,\,to\,\,\phi}{opposite\,\,to\,\,\beta} = \dfrac{DE}{BE} = \dfrac{CE}{AE}$

$DE \times AE = CE \times BE$
 

Also note that
$\varphi = \frac{1}{2}(\theta - \beta)$

Intersecting Tangent and Secant
Tangent EB intersect to secant EC at point E as shown below. Angle BCE is equal to angle ABE, both are denoted by β.
 
Intersecting secant and tangent

 

Triangle ABE is similar to triangle BCE. By ratio and proportion

$\dfrac{opposite\,\,to\,\,\phi}{opposite\,\,to\,\,\beta} = \dfrac{BE}{AE} = \dfrac{CE}{BE}$

$BE^2 = CE \times AE$

 

Video

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