Measurement

Hong Kong

Stage 2

Lesson

You have probably heard of a special number in mathematics called Pi, it has the symbol $\pi$π. But what exactly is it?

Well, Pi is a special number that represents the relationship between any circle's circumference and its diameter.

Let's explore how many diameters fit around the circumference of a circle by using the applet below.

- Change the diameter of the circle using the blue dot.
- Then roll the circle out along the line.
- Once the circle is unravelled you can lay the diameter segments end to end and see how many fit.

You would have found that the length of the circumference was always $3$3 and a bit diameters. In fact, this value is ALWAYS the same, and we call this constant Pi.

The number Pi is irrational (and so the decimals go on and on forever, without ever repeating), that's why we use the symbol $\pi$π.

$\pi=3.141592653589$π=3.141592653589...

If a number is ir*ratio*nal, then we cannot express it as the *ratio* of two whole numbers. Because Pi is irrational, any calculation we do on a calculator with Pi in its decimal form will be an estimate. A pretty close one, but still not what we would call exact. When we want to write an expression in "exact form", we can write it in terms of some multiple of $\pi$π.

It doesn't matter what the diameter is - the relationship between the circumference and diameter is always the same.

$\pi=\frac{C}{d}$π=`C``d`

The circumference of a circle can be calculated with a special formula.

To find the formula for the circumference of a circle we will rearrange the ratio we used to find $\pi$π and make $C$`C` the subject.

Circumference of a circle

$C=\pi d$`C`=π`d`

and because the diameter is twice the radius, we can also write the formula as

$C=2\pi r$`C`=2π`r`

If the diameter of a circle is $7$7 cm, find its circumference to the nearest integer using the approximation $\pi=\frac{22}{7}$π=227

If the radius of a circle is equal to $27$27 cm, find its circumference correct to one decimal place.

Calculate the total perimeter of the following figure, correct to one decimal place.

We have the equation $C=2\pi r$`C`=2π`r` which gives the circumference of a circle in terms of its radius. Let's now rearrange this equation to make the radius the subject.

Dividing both sides of the equation by $2\pi$2π, we are left with

$r=\frac{C}{2\pi}$`r`=`C`2π.

So, substituting the circumference $C$`C` into the equation $r=\frac{C}{2\pi}$`r`=`C`2π will give the exact value for the radius $r$`r`.

Another related part of the circle is the diameter $D$`D`, which is twice as long as the radius $r$`r`. That is,

$D=2r$`D`=2`r`.

This means that we can also find the diameter of a circle when we know its circumference. Since we know that the radius is given by the equation

$r=\frac{C}{2\pi}$`r`=`C`2π,

and that the diameter is twice as long as the radius, this means that

$D=2\times\frac{C}{2\pi}$`D`=2×`C`2π.

Cancelling the common factor of $2$2 between the numerator and denominator, we have that

$D=\frac{C}{\pi}$`D`=`C`π.

So, substituting the circumference $C$`C` into the equation $D=\frac{C}{\pi}$`D`=`C`π will give the exact value for the diameter $D$`D`.

Radius of a circle

If we know the circumference $C$`C` of a circle, we can find its radius $r$`r` by using the relation

$r=\frac{C}{2\pi}$`r`=`C`2π.

Diameter of a circle

If we know the circumference $C$`C` of a circle, we can find its diameter $D$`D` by using the relation

$D=\frac{C}{\pi}$`D`=`C`π.

It is sometimes more useful to know the radius or diameter of a circle as a decimal, rather than the exact value in terms of $\pi$π. If this is the case, we can use a calculator to evaluate the expression and round the result to the desired number of decimal places.

What is the radius $r$`r` of a circle with the circumference $C=14$`C`=14 cm?

Round your answer to two decimal places.

What is the diameter $D$`D` of a circle with the circumference $C$`C` of length $44$44 m?

Round your answer to two decimal places.

Fractional parts of circles are often referred to as sectors. These are effectively pieces of a circle that have been sliced out from the centre.

examples of sectors |

If we wanted the circumference for just $\frac{1}{4}$14 of a circle, we would find the circumference and divide by $4$4.

If we wanted the circumference for just $\frac{1}{3}$13 of a circle, we would find the circumference and find 1 third of it.

So the process is going to be the same as above, except we will also have to find the fractional part of the circumference value.

Remember!

Make sure you read the question correctly. Is it asking for the circumference part, or the full perimeter of the shape.

If it wants the perimeter, don't forget to add on the radius lengths as well.

We want to find the total perimeter of the shape given below.

What fraction of the complete circle is shown?

What is a quarter of the circumference of a circle with radius $3$3 cm? Give your answer in exact form.

Find the total perimeter of one quarter of a circle with radius $3$3 cm. Give your answer in exact form.

We want to find the total perimeter of the shape given below.

What fraction of the complete circle is shown?

What is a half of the circumference of a circle with radius $7$7 cm? Give your answer in exact form.

Find the total perimeter of a semicircle with radius $7$7 cm. Give your answer in exact form.

If you take a look around, you should be able to spot some objects that have circular features. A clock face, a round plate, or the lid of a jar are some examples of things that have the outline of a circle.

For lots of other common objects there are circles present in different parts of their shape. Some examples include the rim of a cup, or the crust on a pizza.

Once we have found a feature of an object that looks like a circle, we can explore the relationship between the radius, the diameter, and the circumference of that circle. In his way we can understand more about the overall object.

Remember!

The diameter $D$`D` of a circle is equal to twice the radius $r$`r`, so $D=2r$`D`=2`r`.

If we know the diameter of a circle, the circumference $C$`C` is given by the equation $C=\pi D$`C`=π`D`.

If we know the radius of a circle, the circumference is given by the equation $C=2\pi r$`C`=2π`r`.

Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of $5$5 m. What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place.

**Think**: The distance around the outside of a circle is its circumference. We can use the radius of the pool to find its circumference.

**Do**: Using the equation $C=2\pi r$`C`=2π`r`, we substitute $r=5$`r`=5 m.

Now we have $C=2\pi\times5=31.41592$`C`=2π×5=31.41592$...$... m, which rounds to $31.4$31.4 m. So Lisa will walk $31.4$31.4 m around the pool.

**Reflect**: Suppose Lisa's neighbour has a circular pool with twice the circumference. What would be its radius?

We can also use these same ideas about circles to understand the circular motion of objects. Think about the way the end of the second hand moves around a clock, or the way a wheel rolls down a hill.

Harry pushes a hula hoop down a hill, and it rolls all the way to the bottom. The hula hoop has a diameter of $85$85 cm. If the hula hoop rolled a total of $25$25 m, how many revolutions did it make on its way down the hill? Give your answer to two decimal places.

**Think**: The hula hoop will make one full revolution when it rolls a distance equal to its circumference. We can use the diameter to find the circumference, then divide the total distance the hoop rolled by this length to find the number of revolutions.

**Do**: Using the equation $C=\pi D$`C`=π`D`, we find that the circumference is $85\pi$85π cm. So the hoop will roll $85\pi$85π cm in one revolution.

Since the total distance is given in metres, we will convert the circumference to the same units. There are $100$100 cm in $1$1 m, so the circumference $85\pi$85π cm is equal to $\frac{85}{100}\pi=0.85\pi$85100π=0.85π m.

Now we can divide the total distance rolled by this circumference.

Number of revolutions | $=$= | $\frac{\text{total distance rolled }}{\text{circumference }}$total distance rolled circumference |

$=$= | $\frac{25}{0.85\pi}$250.85π | |

$=$= | $9.36$9.36 (to two d.p.) |

The hula hoop has rolled about $9.36$9.36 times as it covers the distance of $25$25 m.

**Reflect**: How far would the hula hoop have rolled if it completed exactly $11$11 revolutions?

A scooter tyre has a diameter of $34$34 cm.

Determine the circumference of the tyre, correct to one decimal place.

A coin has a diameter of $2.2$2.2 cm. If the coin is rolled through $45$45 complete revolutions, what distance will it travel?

Give your answer correct to one decimal place.

Carl is performing an experiment by spinning a metal weight around on the end of a nylon thread. How far does the metal weight travel if it completes $40$40 revolutions on the end of a $0.65$0.65 m thread?

Give your answer correct to one decimal place.