MathematicsChapter 11: Practical Geometry of Circles
Completing A Circle Without the Centre (named)
One school method steps congruent triangles / equal chords around the arc until it closes.
Learning objectives
- Explain completing a circle without the centre (named) using the correct definitions, conditions, and notation.
- Apply equal chords around, hexagon 120 deg method correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.
Formula
angle =
Where / symbols
- COMPLETING A CIRCLE WITHOUT THE CENTRE (NAMED)
- One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.
- CONGRUENT TRIANGLES
- Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.
- 120 DEG / HEXAGON
- Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).
Overview
Equal steps on the rim:
Congruent Triangles
Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.
Examples Copy chord length · Step around · Not random radii
120 Deg / Hexagon
Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).
Examples Regular hexagon · Six equal chords · Central angle of hexagon is 60
WHAT — COMPLETING A CIRCLE WITHOUT THE CENTRE (NAMED)
One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Completing A Circle Without The Centre (Named) matters because accurate compass-and-straightedge constructions apply circle theorems in a visible way. Each construction works because a locus or perpendicular condition fixes the required centre, circle, or tangent.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through CONGRUENT TRIANGLES, then 120 DEG / HEXAGON. Worked: interior angle of a regular hexagon used in the 120 deg method. regular hexagon has 6 sides interior angle = [(6-2)*180]/6 = 7206 = 120 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- interior angle of a regular hexagon used in the 120 deg method.
- regular hexagon has 6 sides
- interior angle = [(6-2)*180]/6
- = 7206
- = 120 deg
In simple words
60 deg is the angle at the centre between adjacent hexagon radii, not the interior angle of the hexagon.
Key points
- equal chords around
- hexagon 120 deg method
- equal chords
- interior 120
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Completing A Circle Without the Centre (named)
Sticky Notes · Quick Revision
Core Idea
- One school method steps congruent triangles / equal chords around the arc until it closes. Another uses 120 deg interior angles so a regular hexagon appears (equal chords). These are named methods in the practical-geometry chapter: equal chords, not a freehand guess.
CONGRUENT TRIANGLES
- Copy a chord length repeatedly along the remaining gap. Congruent triangles keep the same radius without naming O at first. Join the new vertices to close the circumference.
120 DEG / HEXAGON
- Regular hexagon interior 120 deg. Building with that angle and equal sides completes the circle in six equal chords. Interior is 120, not 60 (60 is the central angle of the hexagon).
Worked Method
- Worked: interior angle of a regular hexagon used in the 120 deg method.
- regular hexagon has 6 sides
- interior angle = [(6-2)*180]/6
- = 7206
- = 120 deg
Common Misconception
- 60 deg is the angle at the centre between adjacent hexagon radii, not the interior angle of the hexagon.
FBISE Exam Check
- 60 deg is the angle at the centre between adjacent hexagon radii, not the interior angle of the hexagon.
60-second revision
- One school method steps congruent triangles / equal chords around the arc until it closes.
- CONGRUENT TRIANGLES: Copy a chord length repeatedly along the remaining gap.
- 120 DEG / HEXAGON: Regular hexagon interior 120 deg.
- Worked-method reminder: = 120 deg
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