Every point of a circle is the same distance from its centre: the radius. The diameter goes right across through the centre, two radii long (d = 2r). A chord joins any two points; the circumference is the distance all the way round.
Wheels, plates, clocks and pizza: why does a wheel roll smoothly? Because every point on its edge is the same distance from the axle.
A circle is every point the same distance from one point, its centre. That distance is the radius (r).
The diameter (d) goes right across through the centre. It is two radii: d=2r, so r=d÷2.
A chord is a straight line joining any two points on the circle. The diameter is the longest chord.
The circumference (C) is the distance all the way round: the circle's perimeter.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Radius 5
Picture a radius, then another one pointing the opposite way.
TryTry it on the board and compare the result with your prediction.
Diameter 10
Watch the radius and the diameter as the circle grows.
Try it yourselfYour edits are saved in this browser.
Make a circle with diameter 10 cm. What is its radius? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
The parts
Here are the relationships.
Worked exampleBefore each step, say what you think comes next.
d=2rThe diameter is two radii, end to end through the centre.
r=5⇒d=2×5=10Radius to diameter: double.
d=120⇒r=120÷2=60Diameter to radius: halve.
CThe circumference is the distance round; the next lessons find it from d.
Now match the parts.
Read Jay's working. Which part goes right across?
Match each part of a circle to what it is.
radius
diameter
chord
circumference
Jay works out how far a chair at the edge of a round table is from its centre. Click the line where he goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
The radius is half of 7.
Try it yourselfYour edits are saved in this browser.
Make a circle with diameter 7 cm.
Use the board. This ticks when you get there.
A bike wheel has diameter 18 inches. What is its radius, in inches?
A circle has radius 4.5 cm. What is its diameter, in cm?
PractiseA few on your own, to check you have it.
Circle parts
A clock face has radius 12 cm. What is its diameter, in cm?
A coin has diameter 3 cm. What is its radius, in cm?
Review: a box is 2 cm by 3 cm by 4 cm. What is its volume, in cm³?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
The biggest circle in a box
Then: how many of those circles fit side by side in the lid?
Need a hint?
The circle must fit across the narrow way.
Another hint
Its diameter can be at most the shorter side.
A box lid is a rectangle 10 cm by 6 cm. What is the radius of the biggest circle that fits flat inside it? Write your reasons.
Your working
A model solution
The diameter can be at most 6 cm the circle must fit across the shorter side
r=6÷2=3 cm the radius is half the diameter
Check your own working
For teachers
Students name the parts of a circle (centre, radius, diameter, chord, circumference), use d = 2r in both directions, and reason about the biggest circle that fits a rectangle (AC9M7M03, node 1). About 35 minutes.
I can find the diameter from the radius, and the radius from the diameter.
I can explain why the diameter is the longest chord.
Before class
Do the Run board yourself first. Students meet these words informally in primary school; this lesson makes them exact.
The one idea: the radius is centre to edge; the diameter is two radii right across; d = 2r. The common mistake is mixing them up in context (Jay's mistake). Circumference is named here and found in the next two lessons.
Running the lesson
Predict: students who chose 2.5 halved the radius.
Run: ask what stays true as the circle grows.
Investigate: draw a chord near the edge and one through the centre; compare lengths.
Modify: for diameter 7, ask why the radius is not a whole number.
Make: let students sketch the lid before answering.
The hard part (on purpose)
Keeping radius and diameter apart in word problems. Ask students to sketch the circle and mark the given length on it before they double or halve.
When students get stuck
A student calls the diameter the radius (Jay's mistake). Ask: Does it stop at the centre or go right across? Fix: Centre to edge is the radius; edge to edge through the centre is the diameter.
A student doubles when they should halve. Ask: Is the diameter bigger or smaller than the radius? Fix: The diameter is bigger: double the radius; halve the diameter.
A student thinks the circumference is a chord. Ask: Is it straight or curved? Fix: The circumference is the curved distance round.
Stretch and support
Stretch: How many 3 cm radius coasters fit, in a grid, on a 30 cm by 24 cm tray?
Support: Use a paper circle folded in half (the fold is a diameter) and in quarters (the centre).
Where it leads
Next: Discovering π (node 2): how the circumference compares to the diameter.
Where it leads: circle theorems (Year 10), and the area of a circle (Year 8).
What each step is for
radius 5: Committing to a diameter for radius 5 shows whether they know the diameter is two radii, or halve, or mix the two up.
diameter 10: Changing the radius until the diameter is 10 shows the diameter is always twice the radius.
the parts: Each part has its own job; Jay calls the diameter the radius. Every diameter is a chord, but only chords through the centre are diameters.
one change at a time: One change at a time: an odd diameter on the board (a half radius), diameter to radius, radius to diameter with a decimal, then the longest chord and two radii in a line.
circle parts: Doubling, halving and the words, mixed; mixing up radius and diameter is the planted slip; the review keeps volume fresh.
the biggest circle in a box: Fitting the biggest circle in a rectangle means reasoning about the diameter against the shorter side, then halving for the radius.
Now you can
Name the parts of a circle.
Find the diameter from the radius, and the radius from the diameter.