Year 9 Maths The Thai Guy

Lesson 3.1 · 20 min

Inscribed Polygons

Drawing a regular hexagon, square, triangle and octagon inside a circle with only compasses and a straight edge.

In this lesson you will learn to…

  • Know what an inscribed polygon is
  • Construct a regular hexagon and an equilateral triangle using the radius
  • Construct a square and a regular octagon using perpendicular bisectors
  • Explain why each construction works

Why this matters

Designers, architects and artists have used inscribed shapes for thousands of years — in tiled floors, rose windows, logos and Islamic geometric art. They are also a neat way to build shapes exactly, without the small errors that creep in when you measure.

Why the hexagon construction works

Join the centre of a regular hexagon to each vertex. You get six triangles, and the angle at the centre of each is 360∘6=60∘\frac{360^\circ}{6} = 60^\circ. Two sides of each triangle are radii, so they are equal, and the other two angles must also be 60∘60^\circ. The triangles are equilateral — so every side of the hexagon equals the radius.

Building other polygons from these two

PolygonAngle at centreHow to construct it
Equilateral triangle120∘120^\circMake the hexagon’s six points, then join every other point: A, C, E
Square90∘90^\circTwo diameters at right angles (perpendicular bisector)
Regular hexagon60∘60^\circStep the radius round the circle six times
Regular octagon45∘45^\circStart with the square, then bisect each 90∘90^\circ angle at the centre
Regular dodecagon (12 sides)30∘30^\circStart with the hexagon, then bisect each 60∘60^\circ angle

The octagon

After constructing the square AXBY, construct the perpendicular bisectors of AY and AX. They pass through the centre and cut the circle at four new points, halfway between the old ones. The eight points together are the vertices of a regular octagon.

Remember. In a construction, leave all your arcs showing. They are the evidence that you constructed the shape rather than measured it.

Key rules and formulas

  • Inscribed polygon

    A polygon is inscribed in a circle when every one of its vertices lies on the circumference. Every regular polygon can be inscribed in a circle.

  • Construct means compasses and straight edge

    When a question says construct, you may use compasses and a ruler for drawing straight lines — but you must not measure lengths or use a protractor.

  • The hexagon uses the radius

    A regular hexagon is six equilateral triangles meeting at the centre, so each side equals the radius. Step round the circle six times with the compasses set to the radius.

  • Angle at the centre

    Joining the centre to each vertex of a regular -sided polygon splits into equal angles: for a hexagon, for a square, for an octagon.
Try it

Polygons inside a circle

Name
pentagon
Interior angle sum = (n − 2) × 180°
(5 − 2) × 180° = 540°
Each interior angle
108°
Each exterior angle = 360° ÷ n
72°
Angle at the centre = 360° ÷ n
72°

Change the number of sides. The dashed radii split the centre into equal angles of . At 6 sides, each triangle is equilateral — that is why the hexagon side equals the radius.

Worked examples

Worked example 1

An inscribed square

Construct a square inscribed in a circle, using compasses and a straight edge only.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

An inscribed regular hexagon

Construct a regular hexagon inscribed in a circle.

Try it yourself first, then reveal the solution one step at a time.

Practice: inscribed polygons

9 questions

Nine questions. Have compasses and a ruler nearby — trying the constructions on paper makes these much easier.
  1. Question 1 of 9Put in order
    Put the steps for constructing an inscribed regular hexagon in order.

    Put these in the right order. Drag them, or use the arrow buttons.

    1. With the point on A, mark the next arc along and label it B
    2. Draw a circle and keep the compasses set to its radius
    3. Join A to B, B to C, … and F back to A
    4. Mark an arc on the circumference and label it A
    5. Repeat to mark C, D, E and F

  2. Question 2 of 9True or false
    Each side of a regular hexagon inscribed in a circle is the same length as the circle's radius.
  3. Question 3 of 9Click the diagram
    The side of an inscribed regular hexagon is equal to one part of this circle. Click that part.

    Click the right part of the diagram. You can also tab to each part and press Enter.

  4. Question 4 of 9Complete the table
    Complete the angle at the centre for each regular polygon.
    PolygonNumber of sidesAngle at the centre (°)
    Equilateral triangle3
    Square4
    Regular hexagon6
    Regular octagon8
    Regular dodecagon30
  5. Question 5 of 9Explore and match
    Use the explorer to find the regular polygon whose angle at the centre is .

    Move the controls until you match the question, then press Check.

    Name
    pentagon
    Interior angle sum = (n − 2) × 180°
    (5 − 2) × 180° = 540°
    Each interior angle
    108°
    Each exterior angle = 360° ÷ n
    72°
    Angle at the centre = 360° ÷ n
    72°
  6. Question 6 of 9Multiple choice
    You have constructed the six points of an inscribed hexagon, A to F. How do you get an inscribed equilateral triangle?
  7. Question 7 of 9Multiple choice
    Starting from an inscribed square, what do you construct to get a regular octagon?
  8. Question 8 of 9Type the answerChallenge
    A square is inscribed in a circle of radius 10 cm. Its diagonals are diameters. Find the area of the square.
    cm²
  9. Question 9 of 9Choose all that apply
    Which of these are you allowed to use when a question says construct?

    Select every correct answer — there may be more than one.