Math Is Fun Forum

How yo 0lace Pi or any irrational numbers on a graph

Example: Compare a square to a circle of width 3 m
Square's Area = w^2 = 3^2 = 9 m^2

Estimate of Circle's Area = 80% of Square's Area = 80% of 9 = 7.2 m2

Circle's True Area = (Pi/4) × D2 = (Pi/4) × 32 = 7.07 m^22 (to 2 decimals)

The estimate of 7.2 m2 is not far off 7.07 m2

I found this in MIF link : https://www.mathsisfun.com/geometry/circle-area.html

it says area of a circle is = Pi/4 * r^2

but I used to use the formula only Pi * r^2
why the website multiply it by 1/4

I even didn't understand the headline : Comparing a Circle to a Square!!!
it said : A circle has about 80% of the area of a similar-width square.
how did he find that? maybe the circle is 70% of 50%
and how did he say : similar-width! it's first time I read about that term width of a circle

How to calculate a degree of traingle
Sides is 1 length using basic geometry only and Pythagorean theorem without using sine and cosine like we are in 300 BC or 600 BC
I want any simple example
To understand how they were calculate degree in old times
Please this will help me a lot and a lot

The distance between two towns on a map is 5 cm. If the real distance between the two towns is 25 km, what is the scale of the map?

it's in Question5 of this page :

https://www.mathopolis.com/questions/q.html?id=1710&t=mif&qs=1709_1710_1711_1712_1713_1714_1715_1716_3603_3604&site=1&ref=2f6e756d626572732f726174696f2e68746d6c&title=526174696f73#

on MIF

the question is talking about the distance between two city, the answer is B)

the scale of the map is 5 cm : 2,500,000 cm
Which is 1 : 500,000

is the 1 : 500,000 a scale? or a ratio?

at only this question I confused where is the ratio there

it's a scale as in Horse Example :-  https://www.mathsisfun.com/numbers/ratio.html

this horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is

1500 : 2000

What is that ratio when we draw it at 1/10th normal size?

1500 : 2000         = 1500×1/10 : 2000×1/10
         = 150 : 200

so it's a scale, it's the only question that I confused where is the ratio there is a two city on paper they are far 5 cm in real 25 km
where is the the ration there

What is sine and cose
Is sine is a length of something or a degree
I can't imagine the sine how thry measure it
Is it illusion quantity
If sine is a ratio then the degree ia a ratio too
So is sine same as degrees
Or there interactive example of a sine

However if the degree is 60 i can find sin and if aegree is 40 I'll get cos by dividing the opposite by adjucent
Also 90 if wanna find tan

What if the degree is 25 or 170 or 35? What will happen

I have main question about limit
Why archimesdes used two polygon
One inscribed in a circle and second circumscribed around the circle
Why he didn't use only one polygon
Can u interpret this gor me in very easy way pleaee
Because i think it enough to use yhe inscribed polygon to measure the perimeter of the circle

Also archimesdes used " Archimedes used a basic geometry including properties of circle and polygon
ratios of sides of triangles to calculate polygon perimeter (would not have used the word "sine" in his work, as this term was not used until much later in history)"

May i know what is these
1) basics geometry
2) properties of circle abd polygon
3) ratios of sides of traingle
Specially the ratios!! What r these
Is there a book ir websites to illustrate these all of them where can i study these and learn about
Please it's my last question about archimesdes i need fullfill illustratetion in details without using sine and cos and modern calculation method
I want to find Pi just like him and at my home

Archimedes of Syracuse first developed the idea of limits to measure curved figures and the volume of a sphere in the third century b.c. By carving these figures into small pieces that can be approximated, then increasing the number of pieces, the limit of the sum of pieces can give the desired quantity

how was Archimedes  doing that? is there examplew with simple numbers for a simple sphere or any curved figure

what what does " the limit of the sum " mean?

Besides this, there is another rule for accuracy ; that is to say, when an unknown or incommensurable quantity is included between numerical limits not differing by many
units.
Thus if the diameter of a circle contain 497
parts, since it is not possible to ascertain precisely
of how many parts the circumference consists, the
more experienced, in accordance with the views of
Archimedes, have enclosed it within limits, namely
1562 and 1561. Again, if the side of a square
contain 1000 parts, the diagonal will be the square
root of the number 2000000. Since this is an in- commensurable number, we seek for its limits by
extraction of the square root, namely 1415 the
greater limit and 1414 the less limit, or more
accurately 1414(604/2828) greater, and 1414 (604/2829)

I searched a lot but I didn't find these what is 1562 greater limit and 1561 less limit
also what is 1414(604/2828) greater, and 1414 (604/2829)

if the side of square contain 1000 parts the diagonal will be the square root of number 2000?
isn't that should be 1 * square root of 2
why he put six zeros?
that book is talking about accuracy right now
also what is these numbers the 1414? and the 1561

continuous progressions, an arithmetical is one which proceeds by equal intervals

a geometrical, one which advances by unequal and proportionally increasing or
decreasing intervals.

I didn't understand the meaning of that can you please illustrate this in plain English and by numbers or examples?

From web i found : Arithmetic progression is a sequence of numbers where the difference between any two consecutive numbers is the same. For example, series 1,2,3,4 is an arithmetic progression that has a common difference
The difference is 1

I know that but at the top it said, "is one which proceeds by equal intervals."

What is mean these sentence? I don't know what it means by saying proceeds by equal intervals.
From the example above I don't see any intervals I searched a lot I didn't see any arithmetic continuous progressions with intervels only a series and a common difference must of the websites
And does he mean by "proceeds" he mean increasing?
And is arithmetic continuous progressions is the same as Arithmetic sequence? Which I found in https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html