Construct a regular hexagon, triangle and square in a circle
Step a radius-length chord around a circle · Use alternate hexagon vertices for an equilateral triangle · Use perpendicular diameters for an inscribed square
Section Paper 1 · Historical Core · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Perpendicular construction
- Regular polygons
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Step a radius-length chord around a circle
- Use alternate hexagon vertices for an equilateral triangle
- Use perpendicular diameters for an inscribed square
Prepare your paper
When a practice task gives numerical input geometry rather than a printed figure, first prepare that source segment, angle, triangle or circle with measuring tools as needed. Then put the measuring scale or protractor aside for the requested formal copying or bisector steps: use a compass and unmarked straightedge to locate the new points. The preparation supplies the givens; it is not a replacement for the formal construction.
The website diagrams explain relationships; their screen size is not a centimetre calibration or a ready-to-measure template. Keep the input figure and useful construction arcs on paper. Your drawing is self-checked and needs human review for reliable marking.
Hexagon and equilateral triangle
On the supplied circle, choose a starting point A. Set the compass to its radius and step that chord around the circumference, each time choosing the next intersection in the same direction rather than stepping backwards. Six equal chords complete a regular hexagon.
Two radii and each radius-length chord form an equilateral triangle, so each center angle is 60°. Join alternate hexagon vertices A, C and E to form an inscribed equilateral triangle: its three equal arcs each span 120° at the center.
Square from perpendicular diameters
Draw one diameter, construct a perpendicular through the center, and mark that line's two circle intersections. Join the four endpoints in their circular order. Four equal 90° center angles give four equal sides and an inscribed square.
Use compass and unmarked straightedge for formal construction; do not replace the perpendicular by visually guessing 90°. Leave the construction evidence. Numerical measurement can check a drawing afterwards, but does not determine the exact construction.
Sources for this lesson
Pause and explain
Which vertices of an inscribed regular hexagon form an equilateral triangle?
Worked example
Describe an inscribed regular hexagon and equilateral triangle for a circle of radius 3 cm.
Show the worked solution
- Step the 3 cm radius-length chord around the circle to obtain six vertices.
- Join consecutive vertices for the hexagon; each side is 3 cm.
- Join alternate vertices for the equilateral triangle.
Answer Hexagon side 3 cm; alternate vertices form the equilateral triangle.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Construct an inscribed regular hexagon in a radius-4 cm circle. State its side length and each center angle.
Give me a hint
Step the circle radius as a chord.
Compare my reasoning
- Each side equals the 4 cm compass opening.
- Six equal center angles total 360°, giving 60° each.
Side 4 cm; center angles 60°.
Look for these in your work
- Keep the radius opening
- Retain the six locating marks
Construct an inscribed square from a supplied diameter and circle center. Describe the second diameter and vertex order.
Give me a hint
Construct a perpendicular through the center.
Compare my reasoning
- Its two circle intersections supply the remaining vertices.
- Join all four endpoints in circular order, giving 90° gaps at the center.
Use perpendicular diameters; join consecutive endpoints around the circle.
Look for these in your work
- Construct rather than guess perpendicularity
- Use circular vertex order
A radius-3 cm circle is replaced by a radius-6 cm circle. Can the old 3 cm compass span still construct its regular hexagon by radius-chord stepping? Explain.
Give me a hint
The chord must equal the new radius.
Compare my reasoning
- A 3 cm chord is no longer a radius-length chord on the new circle.
- Reset the span to 6 cm so every center triangle is equilateral and subtends 60°.
No. Use a 6 cm compass span for the new hexagon.
Look for these in your work
- Match the current radius
- Justify the 60° center step
Common mistakes
- Choose the forward intersection when stepping the chord.
- An inscribed square's side is not its circle's diameter.
What can you explain now?
Which vertices of an inscribed regular hexagon form an equilateral triangle?
Compare with the explanation
Alternate vertices, such as A, C and E.
The equal skipped arcs give three equal 120° center angles.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Which vertices of an inscribed regular hexagon form an equilateral triangle?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.