Circles

central angle

an angle with its vertex at the center of the circle and with endpoints on the circle’s circumference

Each chord has…

…a corresponding arc

A diameter that is perpendicular to a chord…

…bisects the chord

inscribed angle

an angle formed by 2 chords that have a common endpoint on the circle

minor segment

the smaller region that’s created when a chord divides a circle into 2 parts

major segment

the larger region that’s created when a chord divides a circle into 2 parts

intercepted arc

an arc whose endpoints are located where 2 chords of an inscribed angle intercept the circumference

To describe an intercepted arc

“Angle XYZ is inscribed in arc XZ” or “Angle XYZ intercepts arc XZ”

The measure of an inscribed angle is always…

…half the measure of the arc it intercepts

Central Angle Theorem

the central angle of a circle is twice the measure of an inscribed angle that subtends the same arc

Generally, if a triangle is inscribed in a semicircle…

…it’s a right triangle

If 2 inscribed angles intercept the same arc…

…then they have the same measure

Intersecting Chords Theorem

if 2 chords intersect inside a circle so that one is divided into segments of lengths a and b, and the other is divided into segments of lengths c and d, then ab = cd

The hypotenuse of a right triangle will be…

…a diameter of the triangle’s circumcircle

cyclic quadrilateral

a quadrilateral that can be inscribed in a circle

A quadrilateral is cyclic if and only if…

…its opposite angles are supplementary

tangent line

a line that intersects a circle in exactly one point

point of tangency

the single point where a tangent line touches a circle

The tangent line is perpendicular to the radius…

…at the point of tangency

Generally, if two tangent segments to a circle meet at the same point outside the circle…

…then they are congruent

Two Tangent Theorem

If 2 lines are tangent to a circle from the same external point, the angle between them will be supplementary to the central angle created by the two tangent lines

secant line

a line that intersects a circle in two points

A chord is the segment defined…

…by the two points where a secant line intersects the circle

Two Secants Segments Theorem

S1AX * S1BX = S2AX * S2BX

When to use Two Secants Segments Theorem

When 2 secants intersect outside the circle

To find the angle between two secants that intersect outside the circle

angle(S1BXS2B) = (arc(S1AS2A) - arc(S1BS2B)) / 2

Formula to find a set of vertical angles when two secants intersect within a circle

(arc(S1AS2A) + arc(S1BS2B)) / 2 = angle(S1AXS2A) = angle(S1BXS2B)

Formula to find the measure of the angle created when a secant and tangent intersect on the circle

angle(SASBT) = arc(SASB) / 2

When to use secant-tangent rule

When a secant and tangent intersect outside the circle

secant-tangent rule

S1X * S2X = XT²

radian

a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius

1 radian

the angle measure of the arc that has the same length as the circle’s radius

Equation relating a full circle and radians

2pi radians = 360 degrees

To convert radians to degrees

Multiply the value in radians by 180/pi

To convert degrees to radians

Multiply the value in degrees by pi/180

If a given angle has no units…

…it’s assumed to be in radians

Arc length formula, r is the radius and theta is the angle measure in radians

s = theta * r

Sector area formula, r is the radius and theta is the angle measure in radians

A = (r² * theta) / 2

apothem

the distance from the center of a regular polygon to one of its sides at a right angle

Formula for area of a regular polygon, a being the apothem and P being the perimeter

A = ½aP

Formula for perimeter of a regular polygon, n being the number of sides and s being the side length

P = ns