Circles & polygons
🎯 In this topic you will
- Name the parts of a circle
- Identify, describe, and sketch regular polygons
🧠 Key Words
- chord
- irregular polygon
- polygon
- regular polygon
- sketch
- scalene triangle
- tangent
Show Definitions
- chord: A straight line joining two points on the circumference of a circle.
- irregular polygon: A polygon with sides and angles that are not all equal.
- polygon: A closed, flat shape with three or more straight sides.
- regular polygon: A polygon in which all sides and all interior angles are equal.
- sketch: A quick, freehand drawing that shows the main features of a shape.
- scalene triangle: A triangle with all three sides of different lengths.
- tangent: A straight line that touches a circle at exactly one point without crossing it.
🟢 Circle basics
You already know these parts of a circle.
➕ More parts of a circle
There are two other parts of a circle that you must know.
📏 The chord
A chord of a circle is a straight line that starts and finishes on the circumference of the circle. If the chord passes through the centre of the circle, it is called the diameter.
📐 The tangent
A tangent to a circle is a straight line that touches the circumference of the circle at only one point.
🔷 Regular vs irregular polygons
You already know the difference between a regular polygon and an irregular polygon.
📐 Properties of a regular polygon
You must be able to describe the properties of a regular polygon, as well as sketch the polygon.
⭐ Example: regular pentagon
For example, a regular pentagon has:
- five sides the same length
- five angles the same size
- five lines of symmetry
- rotational symmetry of order $5$
❓ EXERCISES
1. For the diagram shown, write down the letters of the lines that are:
a. tangents to the circle
b. chords of the circle
👀 Show answer
a. $AB$, $BG$, $AG$
These lines touch the circle at only one point each, so they are tangents to the circle.
b. $C$, $D$, $E$, $F$
These are straight lines with both endpoints on the circumference, so they are chords of the circle.
❓ EXERCISES
2. This is part of Omar’s homework.
Label the parts of the circle.
Omar has made a lot of mistakes.
Copy the diagram and correctly label the parts of the circle.
👀 Show answer
The correct labels for the circle are:
- Centre: the red dot in the middle of the circle.
- Radius: a line from the centre to the circumference.
- Diameter: a line passing through the centre with endpoints on the circle.
- Chord: a straight line connecting two points on the circumference (not necessarily through the centre).
- Circumference: the boundary of the circle.
- Tangent: a straight line that touches the circle at exactly one point.
Omar’s diagram included several incorrect placements. The corrected diagram should clearly show each part in the right position.
🧠 Think like a Mathematician
Investigation: Explore the relationship between a tangent and a radius of a circle.
Method:
- Use a pair of compasses to draw four circles of different sizes.
- On each circle, draw a tangent to the circle.
- At the point where your tangent meets your circle, draw a radius to that point. (See example diagram.)
- On each of your circles, measure the angle the radius makes with the tangent.
- Record and compare your answers.
Follow-up Questions:
👀 show answer
- 1: In every case, the radius meets the tangent at a right angle.
- 2: The angle between a tangent and a radius is always $90^{\circ}$.
❓ EXERCISES
4. Marcus draws this diagram.
Draw three circles.
a. Inside the first circle draw three chords that make a scalene triangle.
b. Inside the second circle draw four chords that make a rectangle.
c. Inside the third circle draw four chords that make a kite.
👀 Show answer
b. A rectangle can be drawn by choosing four points on the circle that are evenly spaced (cyclic quadrilateral with opposite sides parallel and equal).
c. A kite is made by drawing four chords where two pairs of adjacent sides are equal in length.
🧠 Think like a Mathematician
Investigation: Drawing chords inside a circle to form a right-angled triangle.
Method:
- Decide on the best method to use to draw three chords inside a circle that will make a right-angled triangle.
- Individually, draw the diagram you planned in step 1.
- Compare your diagram with others and check if the right angle is formed correctly.
- Examine the longest chord of the circle in your diagram.
Follow-up Questions:
👀 show answer
- a) The best method is to draw a diameter of the circle and then join the ends of the diameter to any other point on the circle. By Thales’ theorem, the angle formed at that point will always be a right angle.
- b) The longest chord of the circle is always the diameter.
❓ EXERCISES
6. Zara draws this diagram.
a. Draw a circle. Draw four tangents to the circle to make a square.
b. Describe the properties that characterise a square.
👀 Show answer
a. Method for a circumscribed square:
- Draw a circle with centre $O$.
- Draw two perpendicular radii (e.g., along the horizontal and vertical through $O$).
- At each intersection point of a radius with the circle, construct the tangent line. Four tangents appear.
- The four tangents meet to form a square around the circle.
b. Properties of a square:
- Four equal sides.
- Four right angles ($90^\circ$ each).
- Opposite sides parallel.
- Diagonals equal, perpendicular, and bisect each other.
- Four lines of symmetry and rotational symmetry of order $4$.
❓ EXERCISES
7. This is what Zara says:
a. Is Zara correct? Draw a diagram to help you explain your answer.
b. Describe the properties that characterise a trapezium.
👀 Show answer
b. A trapezium is a quadrilateral with exactly one pair of parallel sides. The other pair of sides is not parallel.
8.
a. Sketch a regular hexagon.
b. Describe the properties that characterise a regular hexagon.
👀 Show answer
b. A regular hexagon has $6$ sides all the same length, all interior angles equal to $120^\circ$, $6$ lines of symmetry, and rotational symmetry of order $6$.
9. The diagram shows a regular decagon. Copy and complete the properties that characterise a regular decagon.
A regular decagon has:
- $\square$ sides the same length
- $\square$ angles the same size
- $\square$ lines of symmetry
- rotational symmetry of order $\square$
👀 Show answer
10. Yasiru has these cards. The cards have different shapes on them.
a. Classify the cards into groups. You must have at least two groups. You can choose how you organise the shapes, but you must explain why you have put the shapes in these groups.
b. Re-classify the cards into different groups. You must have at least two groups. Explain why you have put the shapes into their new groups.
💡 Tip
You can classify using symmetry properties or lengths of sides or number of equal angles, etc.
👀 Show answer
b. Another classification could be based on symmetry: shapes with lines of symmetry versus shapes without. Explanations should focus on properties like equal sides, equal angles, or symmetry.
❓ EXERCISES
11. This is what Zara says:
a. Sketch a regular hexagon. Use Zara’s method to divide the hexagon into identical triangles. How many identical triangles are there?
b. Without drawing any more shapes, copy and complete this table. Explain how you worked out the answers.
| Name of regular polygon | Number of identical triangles inside |
|---|---|
| pentagon | $5$ |
| hexagon | |
| heptagon (7 sides) | |
| octagon | |
| nonagon (9 sides) | |
| decagon |
c. To convince that your method is correct, draw one of the other regular polygons and divide the shape into identical triangles. How many identical triangles are there?
👀 Show answer
a. A regular hexagon divided by drawing line segments from the centre to each vertex gives $6$ congruent isosceles triangles, so there are $6$ identical triangles.
b. In a regular $n$-gon, joining the centre to all vertices creates $n$ identical central triangles. Hence the completed table is:
| Name of regular polygon | Number of identical triangles inside |
|---|---|
| pentagon | $5$ |
| hexagon | $6$ |
| heptagon (7 sides) | $7$ |
| octagon | $8$ |
| nonagon (9 sides) | $9$ |
| decagon | $10$ |
c. Any regular $n$-gon will divide into $n$ identical triangles by drawing segments from the centre to each vertex (e.g., an octagon gives $8$ triangles). This confirms the rule.