The circumference of a circle

A circle is a closed line all of whose points are at the same distance from one point – the centre ($\displaystyle O$ in the drawing).

• Radius ($\displaystyle r$) – a segment from the centre to a point on the circle.
• Diameter ($\displaystyle d$) – a segment through the centre joining two points on the circle.
• A diameter is made of two radii: $\displaystyle {d}={2r}$.
• The circumference is the length of the line itself – the way an ant walks going once around the circle.

The circumference of every circle is a little more than 3 times its diameter.
Pupils measured the diameter of three round things with a tape measure, and their circumference with a string wound around them:

• In every circle, big or small, the circumference divided by the diameter is the same number.
• It is called pi, and written with the Greek letter $\displaystyle \pi$.
• $\displaystyle {\pi}={3.14159\ldots}$ – the digits after the point never end, and never repeat.
• The table does not give exactly the same number, because the measurements had small errors.
• In calculations we use $\displaystyle \pi\approx3.14$.

The circumference divided by the diameter is $\displaystyle \pi$, so the circumference is $\displaystyle \pi$ times the diameter:

The diameter is twice the radius, so the circumference can be found from the radius too:

A wheel’s radius is 30 cm.
How far does the wheel roll in one turn? And in ten turns?

• In one turn the wheel rolls its circumference: $\displaystyle {C}={2\times3.14\times30}={188.4}$ cm.
• In ten turns: $\displaystyle {10\times188.4}={1884}$ cm – almost 19 metres.

Find the circumference of the circle, in cm ($\displaystyle \pi\approx3.14$).

A round table’s circumference is 314 cm.
What is its diameter? And its radius?

• $\displaystyle {C}={\pi d}$, so $\displaystyle {3.14\cdot d}={314}$.
• We divide both sides by 3.14: $\displaystyle {d}={314:3.14}={100}$ cm.
• The radius is half the diameter: $\displaystyle {r}={50}$ cm.

Solve with an equation ($\displaystyle \pi\approx3.14$).

A window has the shape of a semicircle.
Its diameter – the straight bottom side – is 80 cm.
How long is the window’s frame, the straight side included? ($\displaystyle \pi\approx3.14$)

A bicycle wheel’s radius is 35 cm.
On the way to school the wheel turns 1000 times.
How many metres long is the way? ($\displaystyle \pi\approx3.14$)