Pi and the features of circles
Pi, written π, is one of the most famous numbers in mathematics, and yet it comes from something you can see with a piece of string. Wrap a string around any circular object, then lay it straight and compare it to the distance across the circle. The string is always a little more than three times as long. That simple, unchanging fact is the whole idea behind π.
This year you learn to describe how π connects the parts of a circle: the circumference around the outside, the diameter across the middle, and the radius from the centre to the edge. Getting these relationships clear sets up every circle calculation you will meet later.
Where π comes from
Imagine rolling a wheel along the ground for exactly one full turn. The distance it covers is its circumference. If you measure that distance and compare it to the wheel's diameter, you find the circumference is about 3.14 diameters, every single time. Whether the wheel is tiny or enormous, the ratio never changes, and that constant ratio is the number π.
Because the relationship is fixed, knowing one measurement lets you estimate another. A circle with a diameter of 10 centimetres has a circumference of about 3.14 times 10, roughly 31 centimetres. The number π is slightly more than 3, so a quick sanity check is that the distance around a circle is always a bit more than three times the distance across it.
Naming the parts of a circle
Three words describe the key measurements. The radius runs from the centre to the edge. The diameter goes all the way across through the centre, so it is exactly twice the radius. The circumference is the curved distance around the whole circle. Keeping these straight is half the battle, because most mistakes come from mixing up radius and diameter.
The link between them is worth saying plainly: diameter equals two radii, and circumference is about π times the diameter. From those two relationships you can move between any of the measurements. If a problem gives you the radius, double it to reach the diameter, then multiply by π for the circumference. Reading a circle this way turns a daunting shape into a few clear, connected facts.
Teaching tip: the string-and-ruler experiment is worth doing physically. Have the student measure the circumference and diameter of three different round objects, a cup, a tin, a plate, and divide one by the other each time. Getting roughly 3.1 every time, from objects of totally different sizes, is a small revelation and makes π feel discovered rather than handed down.
The most common error is using the radius where the diameter belongs, or vice versa. Encourage a habit of labelling which one a question gives before doing any arithmetic, since the doubling or halving step is where marks are usually lost.
Putting π to work
Once you know the diameter, π turns it into a circumference. The rule is C = π × d, and using 3.14 for π gives a quick estimate. Changing the diameter shows the circumference rising in step, always about three and a bit times as long.
Radius and diameter
The radius and diameter are tied together simply: the diameter is two radii. Laying two equal radii end to end through the centre spans the whole circle. This is why you double a radius to reach the diameter, and halve a diameter to reach the radius.
Why the ratio never changes
The heart of π is that the circumference divided by the diameter is the same for every circle. Two circles of very different sizes both give about 3.14 when you divide one by the other. That unchanging ratio is exactly what the number π names.