Circle Theorems and Formulas

ARC & ANGLE MEASURES IN CIRCLES

Central Angles

• A central angle is an angle whose vertex is at the center of the circle.
• The measure of a central angle is equal to the measure of its intercepted arc.
• Example: If m\angle ACB = 140^\circ, then m\stackrel\frown{AB} = 140^\circ.

Inscribed Angles

• An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.
• The measure of an inscribed angle is half the measure of its intercepted arc.
• m\angle ABC = \frac{1}{2} m\stackrel\frown{AC}

Overlapping Arcs

• m\angle ABD = m\angle ACD

Intersecting Chords or Secants (on the Interior)

• If two chords or secants intersect inside a circle, the measure of the angle formed is half the sum of the measures of the intercepted arcs.
• m\angle 2 = \frac{1}{2} (m\stackrel\frown{AB} + m\stackrel\frown{CD})

Intersecting Secants (on the Exterior)

• If two secants intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.
• m\angle A = \frac{1}{2} (m\stackrel\frown{BD} - m\stackrel\frown{BC})

Intersecting Secants & Tangents (on the Exterior)

• If a secant and a tangent intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.
• m\angle A = \frac{1}{2} (m\stackrel\frown{BD} - m\stackrel\frown{BC})

Intersecting Tangents & Chords/Secants (on the circle)

• m\angle 1 = \frac{1}{2} (m\stackrel\frown{ABC} - m\stackrel\frown{AC})
• m\angle 2 = \frac{1}{2} (m\stackrel\frown{AC})

Intersecting Tangents (on the Exterior)

• m\angle A = \frac{1}{2} (m\stackrel\frown{DE} - m\stackrel\frown{BC})

Inscribed Quadrilaterals

• If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
• m\angle B + m\angle C = 180^\circ
• m\angle A + m\angle D = 180^\circ

SEGMENT LENGTHS IN CIRCLES

Tangents

• If a line (AB) is tangent to a circle (C) at a point, then the tangent line is perpendicular to the radius (CD) drawn to that point.
• AB \perp CD

Two Tangents from the same External Point

• If two tangent segments are drawn to a circle from the same external point, then the tangent segments are congruent.
• If AB and BC are tangent to circle D, then AB = BC.

Intersecting Chords (on the Interior)

• If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
• a \cdot b = c \cdot d

Intersecting Secants (on the Exterior)

• If two secant segments are drawn to a circle from the same external point, then the product of the length of one secant segment and its external segment equals the product of the length of the other secant segment and its external segment.
• a(a+b) = c(c+d)

Intersecting Tangent & Secant (on the Exterior)

• If a tangent segment and a secant segment are drawn to a circle from the same external point, then the square of the length of the tangent segment is equal to the product of the length of the secant segment and its external segment.
• a^2 = b(b+c)

CIRCUMFERENCE, ARC LENGTH, & CIRCLE EQUATIONS

Circumference

• Circumference of a circle is the distance around the circle.
• C = 2\pi r
• C = \pi d

Area

• Area of a circle is the region enclosed by the circle.
• A = \pi r^2

Arc Length

• Arc length is the distance along an arc of a circle.
• Arc Length = x \cdot 2\pi r

Standard Equation of a Circle

• The standard equation of a circle with center (h, k) and radius r is:
• (x-h)^2 + (y-k)^2 = r^2
• (h, k) is the center and r is the radius