Second Semester Geometry

Survival Guide

Unit 9: Circles

Big Ideas



  • Defined as the set of points in a plane that are equidistant from a fixed point called the center
  • Is named by the center
  • Has three regions: Interior/Exterior/On

  • A segment whose endpoints are the center of a circle and a point on the circle
  • All radii on the same circle are congruent
  • labeled as r

  • A segment whose endpoints are two points on a circle

  • A chord that passes through the center
  • 2r

  • Distance around a circle
  • 2πr or πd

  • πr²

-Concentric Circles
  • Two circles that share a common center

-Proving Similarity
  • Only a translation and dilation are necessary



  • A line in the plane of a circle that intersects the circle in exactly one point
  • A tangent line is // to the r
  • Tangents from a common external point are congruent


  • Part of a circle
  • Congruent arcs: Two equal arcs of the same circle or congruent circles are congruent.
  • Minor arc:<180°
  • Major Arc:>180°
  • Semicircle:=180°


-Inscribed Angle

  • An angle whose vertex is on a circle and whose sides contain chords of the circle
  • The measure of an inscribed angle is half the intercepted arc: angle=½arc
  • If two inscribed angles intercept the same arc, then the angles are congruent
  • A polygon is and inscribed polygon if all of its vertices lie on a circle, the circle that contains the vertices is a circumscribed circle
  • If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle
  • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary


-Tangent Angle

  • If a tangent and a cord intersect at a point on a circle, then the measure of each angle formed is half the measure of its intercepted arc
  • Angle=½arc
-Interior Angle
  • If two cords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle
  • Angle=½(arc1+arc2)

-Exterior Tangent/Secant Angle

  • If 2 tangents, 2 secants, or a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs
  • Angle=½(arc1-arc2)


-Arc Length Corollary

  • In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°
  • Arc Length=arc/360°(2πr)


-Areas of Sectors

  • region bounded by two radii of the circle and their intercepted arc
  • The ratio of the area of a sector of a circle is equal to the ratio of the measure of the intercepted arc to 360°
  • Sector Area=arc/360°(πr²)

Tips to students

  • Remember how ratios work
  • Remember formulas for circumference and area
  • Remember how to find angle measures with the different formulas

Things I Struggled With

  • Remembering the formulas for finding the angle
  • I kept practicing it till I could remember.