The circumference and π

Measures and solids, grade 6 Lesson 9 of 16Learning

The circumference and π

Measuring the circumference

The circumference of a circle is the length of its line – the distance an ant walks going once around the circle.

It is measured with a string: wrap a string around a round object, mark where it meets itself, then straighten it and measure it with a ruler.
For a more exact measure, wrap the string around 10 times and divide the length by 10.

Pupils measured the diameter and the circumference of four round objects this way:

Object	Diameter (cm)	Circumference (cm)
a cup	8	25
a plate	24	75.6
a bicycle wheel	50	157
a hoop	80	252

Exercise 1

How many times the diameter is the circumference?
For each object in the table, divide the circumference by the diameter.

a cup

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a plate

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a bicycle wheel

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a hoop

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The number π

In every circle, big or small, the circumference is the same number of times the diameter – a bit more than 3.

This number is called pi, and it is written with the Greek letter $\displaystyle \pi$ .
$\displaystyle {\pi}={3.14159\ldots}$ – its digits after the point never end and never repeat.
In calculations we use a number close to it: $\displaystyle \pi\approx3.14$ . The sign $\displaystyle \approx$ means “is about equal to”.
The measurements gave slightly different numbers, because measuring with a string and a ruler is never quite exact.
The Bible (1 Kings 7) describes a round bronze basin, “the Sea”, 10 cubits across and 30 cubits around: 3 times – an approximation of $\displaystyle \pi$ .

The circumference formula

$\displaystyle C$ – the circumference
$\displaystyle d$ – the diameter
$\displaystyle r$ – the radius

The circumference is $\displaystyle \pi$ times the diameter:

\displaystyle C

\displaystyle =

\displaystyle \pi d

A formula writes numbers and letters side by side, with no times sign: $\displaystyle \pi d$ means $\displaystyle \pi\times d$ .

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

\displaystyle C

\displaystyle =

\displaystyle 2\pi r

$\displaystyle 2\pi r$ is about $\displaystyle 6.28\times r$ .

Finding the circumference

The radius of a wheel of Noa’s bicycle is 30 cm.
How many cm does the wheel go in one turn? And in ten turns?

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

Exercise 2

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

the diameter is 10 cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

the radius is 7 cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a 10-shekel coin: the diameter is 2.3 cm

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A fence is built around a round flower bed whose radius is 2 m.
How long is the fence, in metres? ( $\displaystyle \pi\approx3.14$ )

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A window is shaped like half a disc.
Its diameter – the straight side at the bottom – is 80 cm.
How long is the window’s frame, the straight side included? ( $\displaystyle \pi\approx3.14$ )

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

From the circumference to the diameter

When the circumference is known, the diameter is found by dividing: the circumference is 3.14 times the diameter, so the diameter is the circumference divided by 3.14.
For example, a round table’s circumference is 314 cm:

The diameter: $\displaystyle {314:3.14}={100}$ cm.
The radius is half the diameter: 50 cm.

Exercise 5

A round cake’s circumference is 62.8 cm.
What is its diameter? And its radius? ( $\displaystyle \pi\approx3.14$ )

The diameter

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The radius

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 6

The metre was fixed so that the equator – the biggest circle around the Earth – would be 40,000 km long.
About how long is the Earth’s radius?
Round the answer to the nearest hundred kilometres.

Formulas: measures and solids, grade 6

Open as a page

The circle and the disc

The diameter $\displaystyle d$ is two radii $\displaystyle r$
$\displaystyle d$ $\displaystyle =$ $\displaystyle 2\times r$
The number pi – the value to calculate with
$\displaystyle \pi\approx3.14$
The circumference of a circle – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference of a circle – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$

Volumes

The volume of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\times b\times c$
The volume of a box, a prism or a cylinder – $\displaystyle S$ the area of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\times h$
The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2 h$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\times h}{3}$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2 h}{3}$
The volume of a sphere – two thirds of the cylinder around it
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{2}{3}\times\pi r^2\times2r$