MathematicsChapter 9: Chord and Arcs of a Circle
Minor Arc, Major Arc, Semicircle
Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way).
Learning objectives
- Explain minor arc, major arc, semicircle using the correct definitions, conditions, and notation.
- Apply minor = shorter, semicircle on a diameter correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way). A semicircle is half the circumference, sitting on a diameter. Equal central angles (same circle) make equal arcs. Arc measure in degrees equals the central angle in the school treatment.
Mathematical form
AOB = 80 deg. Find the major arc AB in degrees
AB = central angle
circle = 360 deg major arc AB
AB = 280 deg
Where / symbols
- MINOR ARC, MAJOR ARC, SEMICIRCLE
- Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way). A semicircle is half the circumference, sitting on a diameter. Equal central angles (same circle) make equal arcs. Arc measure in degrees equals the central angle in the school treatment.
- MINOR AND MAJOR
- Unless AB is a diameter, one arc is minor and one is major. Naming arc AB without 'minor/major' can be ambiguous; many papers mean the minor. Equal chords sit on equal minor arcs.
- SEMICIRCLE
- Central angle 180 deg. The bounding chord is a diameter. Angle at the circumference in that semicircle is 90 deg (Chapter 10). Arc length school formula uses fraction of circumference if asked: (theta/360)*2 pi r.
Relationship
- minor = shorter
Overview
Short way, long way, half:
Minor and Major
Unless AB is a diameter, one arc is minor and one is major. Naming arc AB without 'minor/major' can be ambiguous; many papers mean the minor. Equal chords sit on equal minor arcs.
Examples Minor < 180 deg typically · Major the rest of 360 · Equal chords, equal minor arcs
Semicircle
Central angle 180 deg. The bounding chord is a diameter. Angle at the circumference in that semicircle is 90 deg (Chapter 10). Arc length school formula uses fraction of circumference if asked: (theta/360)*2 pi r.
Examples 180 deg at O · Chord is diameter · Rim angle 90
WHAT — MINOR ARC, MAJOR ARC, SEMICIRCLE
Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way). A semicircle is half the circumference, sitting on a diameter. Equal central angles (same circle) make equal arcs. Arc measure in degrees equals the central angle in the school treatment. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Minor Arc, Major Arc, Semicircle matters because chord and arc theorems convert circle diagrams into equal lengths, equal angles, and right triangles. They are the foundation for later tangent proofs and constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through MINOR AND MAJOR, then SEMICIRCLE. Worked: central angle AOB = 80 deg. Find the major arc AB in degrees. minor arc AB = central angle = 80 deg full circle = 360 deg major arc AB = 360 - 80 major arc AB = 280 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- central angle AOB = 80 deg. Find the major arc AB in degrees.
- minor arc AB = central angle = 80 deg
- full circle = 360 deg
- major arc AB = 360 - 80
- major arc AB = 280 deg
In simple words
Arc measure in degrees is not the chord length. An 80 deg arc on a huge circle is still 80 deg but a long curve.
Key points
- minor = shorter
- semicircle on a diameter
- not a diameter => two sizes
- diameter cuts two semicircles
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Minor Arc, Major Arc, Semicircle
Sticky Notes · Quick Revision
Core Idea
- Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way). A semicircle is half the circumference, sitting on a diameter. Equal central angles (same circle) make equal arcs. Arc measure in degrees equals the central angle in the school treatment.
MINOR AND MAJOR
- Unless AB is a diameter, one arc is minor and one is major. Naming arc AB without 'minor/major' can be ambiguous; many papers mean the minor. Equal chords sit on equal minor arcs.
SEMICIRCLE
- Central angle 180 deg. The bounding chord is a diameter. Angle at the circumference in that semicircle is 90 deg (Chapter 10). Arc length school formula uses fraction of circumference if asked: (theta/360)*2 pi r.
Worked Method
- Worked: central angle AOB = 80 deg. Find the major arc AB in degrees.
- minor arc AB = central angle = 80 deg
- full circle = 360 deg
- major arc AB = 360 - 80
- major arc AB = 280 deg
Common Misconception
- Arc measure in degrees is not the chord length. An 80 deg arc on a huge circle is still 80 deg but a long curve.
FBISE Exam Check
- Arc measure in degrees is not the chord length. An 80 deg arc on a huge circle is still 80 deg but a long curve.
60-second revision
- Two points A and B on a circle cut two arcs: the minor (shorter, less than a semicircle) and the major (the long way).
- MINOR AND MAJOR: Unless AB is a diameter, one arc is minor and one is major.
- SEMICIRCLE: Central angle 180 deg.
- Worked-method reminder: major arc AB = 280 deg
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