Tenth Grade · Math · week 31

Circles: central angles, arcs, and chords

Apply the central angle/arc relationship, the Arc Addition Postulate, and the congruent-chords-congruent-arcs relationship.

A circle is the set of all points a fixed distance (the radius) from a center point; a diameter is a chord through the center, twice the radius. A central angle has its vertex at the center, with two radii as sides; the Arc Addition Postulate works just like segment and angle addition -- the measure of a central angle EQUALS the measure of its intercepted arc, the piece of the circle it "cuts off." A minor arc (less than 180 degrees) is named with its two endpoints; a major arc (more than 180 degrees) is named with three points to avoid ambiguity; an arc of exactly 180 degrees is a semicircle.

Chords and arcs are linked tightly: in the same circle (or congruent circles), congruent chords cut off congruent arcs, and congruent arcs come from congruent chords -- the relationship runs both ways. One more fact is worth knowing by name: a diameter (or any radius) perpendicular to a chord always bisects that chord AND bisects its arc, splitting both into two congruent halves -- a fact that comes from the two halves being mirror images across that diameter.

📋 For the teacher
Teach it (10–15 min):
  • Use an actual circular object (a plate, a clock face) and a piece of string to physically show a central angle and the arc it "cuts off" -- the equal-measure fact should feel obvious once seen this way.
  • Practice naming arcs correctly (two letters for minor, three for major) with several examples, since ambiguous naming is a common early mistake.
  • Fold a paper circle along a diameter perpendicular to a chord to physically show the chord (and its arc) landing exactly on top of itself.
  • Keep "the whole circle is 360 degrees" posted as a constant anchor fact, the same way "a triangle is 180 degrees" anchored the triangle unit.
Talk about it:
  • Why does a central angle always equal the measure of its intercepted arc?
  • Why does naming a major arc require three points instead of two?
  • Why does a chord perpendicular to a diameter get bisected by it?

Mastery looks like: They apply the central angle/arc relationship, the Arc Addition Postulate, the congruent-chords-congruent-arcs relationship, and the perpendicular diameter/chord bisection theorem to find unknown arc and chord measures.

Common stumbles: Forgetting the whole circle is 360 degrees when finding a major arc or a third central angle; naming a major arc with only two letters, creating ambiguity.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

AdditionSee groups combine — with counters, blocks, and the number line.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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