How to properly calculate long calculations

What is pi ($$ pi $$)?

Find some round objects. Measure the diameter and the circumference of the item.

You can measure the circumference by putting a thread around the item and then measuring the length of the thread.

Then calculate the ratio of circumference and diameter, i.e. $$ u / d $$.

Already in the Bible there is evidence that people have studied the relationship between circumference and radius.
The Greek mathematician Archimedes succeeded around 250 BC. BC for the first time to limit this relationship mathematically.
Today we know that the ratio of circumference to diameter is always the same $$ pi $$.

What is pi ($$ pi $$)?

Here are Christian and Tamara's results.

Scope $$ u $$ Diameter $$ d $$ $$ u / d $$
cups 27 cm 8.5 cm 3,18
Bangle 18 cm 5.8 cm 3,10
Pencil Case 21 cm 6.5 cm 3,23
ball 100 cm 32 cm 3,13

The ratio of circumference to diameter is always the same as $$ pi $$.

$$ pi $$ is an irrational number with an infinite number of decimal places.

$$ pi $$ = 3.141592654….

Christian and Tamara are already very close to the correct result with their calculations. The deviations result from measurement inaccuracies.

$$ pi $$ is not a rational number. This means that it cannot be represented as a fraction and has an infinite number of places after the decimal point.
$$ pi $$ $$ approx $$ 3.14

Lines in a circle

In memory of:

The diameter is twice the radius.

$$ d = 2 * r $$

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The circumference formula

If you divide the circumference (u) by the diameter (d) of any circle, you always get the number $$ pi $$.

So the following applies:

$$ pi = u / d $$

This gives you the circumference formula for the circle.

$$ u = pi * d $$

or because of $$ d = 2 * r $$

$$ u = 2 * pi * r $$

Note: If you don't have a calculator with the $$ pi $$ key, use $$ pi approx 3.14 $$.

$$ pi = $$ circumference (u) divided by diameter (d)

$$ pi = u / d rArr u = pi * d $$

or

$$ u = 2 * pi * r $$

Calculation of the circumference for a given diameter

How far does a wheel with the diameter $$ d = 70 $$ cm roll with exactly one revolution?

With exactly one revolution, a wheel covers exactly the distance of the circumference. Just apply the formula.

$$ u = pi * d $$

$$ u = pi * 70 $$ cm

$$ u approx 219.9 $$ cm $$ approx $$ $$ 2.2 $$ m

With one full revolution, the wheel covers about $ 2.2 $ m.

$$ u = pi * d $$

$$ u = 2 * pi * r $$

Calculation of the circumference for a given radius

The earth has a radius of about $ 6370 $ km.

Calculate the length of the equator, i.e. the circumference of the earth.
(The earth is not exactly a sphere. But you can neglect that.)

$$ u = 2 * pi * r $$

$$ u = 2 * pi * 6370 $$ km

$$ u approx 40,024 $$ km

The length of the equator is approximately $$ 40,024 $$ km.

$$ u = pi * d $$

$$ u = 2 * pi * r $$

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Calculation of the diameter and the radius for a given circumference

You measure the circumference of a glass with $$ 21 $$ cm.

Calculate the diameter and also the radius of the glass.

$$ d = u / pi $$

$$ d = (21 cm) / pi $$

$$ d approx 6.68 $$ cm

The diameter of the glass is approximately $ 6.68 $ cm.

Since you know the diameter is twice the radius, all you have to do is divide the result by two to calculate the radius.

The radius of the glass is therefore approximately $ 3.34 $ cm.

$$ u = pi * d $$

$$ u = 2 * pi * r $$

$$ d = u / pi $$