Two circles have the same center. One has radius 3 cm and the other has radius 6 cm. Each has a central angle of 60°. Which statement is true?
ABoth arcs have the same measure (60°) and the same length
BThe larger circle's arc has a greater degree measure because its radius is larger
CBoth arcs have the same degree measure (60°), but the larger circle's arc is longer
DThe smaller circle's arc is longer because it curves more sharply
Arc measure (in degrees) equals the central angle — 60° in both cases — regardless of radius. But arc length is a linear distance that depends on radius: a larger circle has a longer circumference, so the same 60° represents a longer physical arc. This is the critical distinction between arc measure and arc length. Two arcs can have identical degree measures yet completely different lengths if their circles have different radii.
Question 2 Multiple Choice
In a circle, points A, B, and C lie on the circle with C on the minor arc from A to B. The measure of arc AC is 40° and the measure of arc CB is 75°. What is the measure of central angle AOB (where O is the center)?
A35°
B75°
C115°
D245°
By the Arc Addition Postulate, arc AB = arc AC + arc CB = 40° + 75° = 115°. The central angle AOB equals the measure of its intercepted arc, so the central angle is also 115°. The Arc Addition Postulate works just like the Segment Addition Postulate: if C is between A and B on the arc, the pieces add to the whole.
Question 3 True / False
A central angle of 90° always intercepts an arc of 90°, no matter the size of the circle.
TTrue
FFalse
Answer: True
Arc measure equals the central angle, always — this relationship is independent of the circle's radius. A 90° central angle cuts off exactly one quarter of the circle (since 90/360 = 1/4), so the arc measures 90° whether the circle has radius 2 cm or radius 200 km. The proportional relationship between central angle and arc is what makes arc measure useful: it depends only on the angle, not on the circle's size.
Question 4 True / False
If two arcs have the same degree measure, they is expected to have the same arc length.
TTrue
FFalse
Answer: False
This is the most common misconception about arc measure. Arc measure (degrees) and arc length (a linear distance) are different quantities. Two arcs with the same degree measure have the same arc length only if they belong to circles of the same radius. A 90° arc on a circle of radius 10 cm is much longer than a 90° arc on a circle of radius 1 cm, even though both arcs measure 90°. Arc length formula (which you'll study next) shows that arc length = (θ/360°) × 2πr — the radius r appears in the formula for length but not for measure.
Question 5 Short Answer
A diameter divides a circle into two semicircles. Explain why each semicircle measures exactly 180°, using the relationship between central angles and arc measures.
Think about your answer, then reveal below.
Model answer: A diameter passes through the center, so it forms a central angle of 180° (a straight angle). By the central angle-arc measure relationship, the intercepted arc equals the central angle — so each semicircle measures 180°. Alternatively: a full circle is 360°, and a diameter divides it into two equal arcs (since the diameter is a line of symmetry through the center), so each arc is 360° ÷ 2 = 180°.
Both approaches reach the same answer. The first uses the definition directly: the central angle formed by a diameter is a straight angle (180°), and arc measure equals central angle. The second uses the symmetry of a diameter plus the fact that the full circle is 360°. Understanding why the arc is 180° — not just that it is — requires connecting the straight-angle property of a diameter to the arc measure theorem.