Circles
🎯 In this topic you will
- Identify and name the parts of a circle
- Draw circles accurately
🧠 Key Words
- centre
- circumference
- compasses
- diameter
- radius
Show Definitions
- centre: The fixed point in the middle of a circle, equally distant from every point on the circumference.
- circumference: The curved boundary line that forms the outside edge of a circle.
- compasses: A drawing instrument used to draw circles and arcs accurately.
- diameter: A straight line passing through the centre of a circle and joining two points on the circumference.
- radius: A straight line from the centre of a circle to any point on its circumference.
🔴 Finding the Centre of the Circle
The red dot shows the centre of the circle.
📏 Understanding the Radius
When you draw a circle, you draw a set of points that are the same distance from the centre. This distance is called the radius of the circle.
⭕ What Is the Circumference?
The perimeter of the circle is called the circumference.
📐 Understanding the Diameter
The diameter of the circle is a line joining two points on the circumference that goes through the centre of the circle.
❓ EXERCISES
1. Copy the diagram. Label the parts of the circle shown using the following words: centre, diameter, radius, circumference.
👀 Show answer
The centre is the point in the middle of the circle. The radius is a line segment from the centre to any point on the circle. The diameter is a line segment that passes through the centre and connects two points on the circle. The circumference is the distance around the circle (the circle's perimeter).
2. Draw a circle with a radius of
a. $5$ cm
b. $60$ mm
👀 Show answer
a. Using a compass, set the distance between the point and the pencil to $5$ cm. Place the point at the centre and draw the circle.
b. $60$ mm is equal to $6$ cm. Set the compass to $6$ cm and draw the circle.
🧠 Think like a Mathematician
Question: Arun draws a circle with radius $3$ cm. Sofia draws a circle with diameter $60$ mm. Arun says: "Our circles are exactly the same size!"
Tip: Remember that $1$ cm $= 10$ mm.
Method:
- Convert both measurements to the same unit (either cm or mm) so you can compare them fairly.
- Recall the relationship between the radius and the diameter of a circle.
- Compare the size of Arun's circle to the size of Sofia's circle.
- Write down a general rule connecting the radius and diameter of any circle.
Follow-up Questions:
Show Answers
- a: Yes, Arun is correct. Sofia's circle has a diameter of $60$ mm. Since $1$ cm $= 10$ mm, $60$ mm is equal to $60 \div 10 = 6$ cm. The diameter of a circle is twice its radius, so Sofia's radius is $6 \div 2 = 3$ cm. Both circles have a radius of $3$ cm, so they are the same size.
- b: The diameter of a circle is twice the length of its radius. In symbols: $d = 2r$ or $r = \frac{d}{2}$.
- c: (Discussion completed — your reasoning should confirm that converting to the same unit is essential before comparing measurements, and that the rule $d = 2r$ holds for all circles.)
❓ EXERCISES
3. These cards show different measurements.
- A Radius = $2$ cm
- B Radius = $10$ cm
- C Diameter = $8$ cm
- D Diameter = $20$ cm
- E Radius = $4$ cm
- F Radius = $20$ mm
- G Radius = $40$ mm
- H Diameter = $4$ cm
- I Diameter = $200$ mm
Sort the cards into groups of measurements that will give the same size circles.
👀 Show answer
Group 1 (radius $2$ cm): A, H (diameter $4$ cm = radius $2$ cm)
Group 2 (radius $4$ cm): C (diameter $8$ cm = radius $4$ cm), E, G (radius $40$ mm = $4$ cm), I (diameter $200$ mm = $20$ cm diameter? Wait, check: $200$ mm = $20$ cm diameter, so radius $10$ cm, not $4$ cm — correction below)
Corrected grouping after unit conversion:
- Radius $2$ cm: A ($2$ cm), H (diameter $4$ cm → radius $2$ cm)
- Radius $4$ cm: C (diameter $8$ cm → radius $4$ cm), E ($4$ cm), G ($40$ mm = $4$ cm)
- Radius $5$ cm? None
- Radius $10$ cm: B ($10$ cm), D (diameter $20$ cm → radius $10$ cm), I (diameter $200$ mm = $20$ cm diameter → radius $10$ cm)
- Radius $1$ cm? None
- F ($20$ mm = $2$ cm radius): So F joins Group 1 (radius $2$ cm) with A and H.
Final groups:
Group 1 (radius $2$ cm): A, F, H
Group 2 (radius $4$ cm): C, E, G
Group 3 (radius $10$ cm): B, D, I
4. This is part of Gethin's homework.
Task: draw and label a diameter onto this circle.
a. Explain the mistake that Gethin has made.
b. Draw out a correct solution for him.
👀 Show answer
a.Gethin has drawn a radius (a line from the centre to the circumference) but labelled it as the diameter. The diameter must pass through the centre and touch the circle at two opposite points.
b. Correct solution: Draw a straight line through the centre that meets the circle at two points. Label this line as "diameter".
5.
a. Draw a dot and label the point A. Make sure there is about $4$ cm of space above, below, to the left and to the right of your point.
b. Draw the set of points that are exactly $3.5$ cm from the point A.
🧠 Reasoning Tip
This means draw a circle of radius $3.5$ cm.
👀 Show answer
a. Place a dot on your paper and label it A. Ensure there is enough space (about $4$ cm) around it.
b. Using a compass, set the radius to $3.5$ cm. Place the compass point on A and draw a circle. This circle is the set of all points exactly $3.5$ cm from A.
🧠 Think like a Mathematician
Question: How accurately can you draw circles when given a radius or diameter measurement?
Equipment: Paper, pencil, ruler, compass
Method:
- On a piece of paper, write down:
- i. a radius length between $30$ mm and $80$ mm
- ii. a diameter length between $7$ cm and $15$ cm.
- Swap your piece of paper with a partner and ask them to draw the two circles with the radius and diameter that you have given.
- Swap back pieces of paper and mark each other’s work.
- Discuss with your partner: How accurate were you and how accurate was your partner? If you were not very accurate, discuss ways that you can improve the accuracy of your drawings.
Follow-up Questions:
Show Answers
- a: Example answer: radius $50$ mm, diameter $10$ cm. (Your own numbers may be different.)
- b: To check accuracy, measure the radius or diameter of the drawn circle using a ruler. For the circle drawn from a given radius, measure from the centre to any point on the circle. For the circle drawn from a given diameter, measure the widest distance across the circle through the centre. Compare these measurements to the original numbers.
- c:
- Make sure the compass pencil is sharp to get a fine line.
- Hold the compass at the top, not the legs, to keep the radius fixed.
- Check the compass setting against a ruler before drawing.
- Draw lightly at first, then darken the final circle.
- Keep the compass point firmly in the centre while turning.
- When measuring a diameter, be sure the ruler passes through the centre.
🧠 Think like a Mathematician
Question: What is the relationship between the distance between the centres of two touching circles and their radii?
Equipment: Paper, pencil, ruler, compass
Method:
- Draw a circle with radius $6$ cm. Label the circle A.
- Draw a circle with radius $2.5$ cm, so that it touches circle A. Label the circle B. Your diagram should look something like this:
- With a ruler, accurately measure the distance between the centre of circle A and the centre of circle B.
- What do you notice about your answer to part c and the radii measurements of circles A and B?
- Draw two more circles that touch. Choose your own radii measurements. Measure the distance between the centres of your two circles. What do you notice?
- Copy and complete this general rule:
"The distance between the centres of two touching circles is the same as the ______."
💡 Tip: Radii is the plural of radius. One radius, two radii.
Follow-up Questions:
Show Answers
- a: (Practical task completed on paper. Use compass set to $6$ cm for circle A and $2.5$ cm for circle B, positioned so the circles touch externally.)
- b: The measured distance should be approximately $8.5$ cm (the sum of the radii: $6 + 2.5 = 8.5$ cm).
- c: The distance between centres equals the sum of the two radii.
- d: For any two touching circles (externally), the distance between centres equals the sum of their radii. For example, if you choose radii $3$ cm and $4$ cm, the distance between centres should be $7$ cm.
- e: "The distance between the centres of two touching circles is the same as the sum of their radii." (For circles that touch externally.)
Note: If circles touch internally, the distance between centres equals the difference of the radii. This investigation focuses on external touching.