why is the "diameter ratio" of a circle surrounded by a ring with equal area = sin 45

[gareth]

why is the "diameter ratio" of a circle surrounded by a ring with equal area = sin 45

A circle, say 20cm in diameter has another circle inside it, such that its area is the same as the surrounding rings' area
The ratio of these diameters is 0,7071 which is sin 45 degrees


area of outer circle is pi*r² = 314,1592 cm²

half of this is 157,0796 which is the area of the inner circle

radius of inner circle is square root of 157,0796/pi = 7,0710

diameter of inner circle = 2 * 7,0710 = 14,14213562

Ratio of diameters 20cm to 14.142cm is 0,7071 which is sin 45

How does this ratio occur mathematically for circles of any diameter ?
or what other way can I see it?

 

[Ishuda]

A circle, say 20cm in diameter has another circle inside it, such that its area is the same as the surrounding rings' area
The ratio of these diameters is 0,7071 which is sin 45 degrees


area of outer circle is pi*r² = 314,1592 cm²

half of this is 157,0796 which is the area of the inner circle

radius of inner circle is square root of 157,0796/pi = 7,0710

diameter of inner circle = 2 * 7,0710 = 14,14213562

Ratio of diameters 20cm to 14.142cm is 0,7071 which is sin 45

How does this ratio occur mathematically for circles of any diameter ?
or what other way can I see it?

Your number values do not reflect what you are saying. For example, pi*r² = 314,1592 cm² [or 314.1592 cm² as I would be inclined to write it] implies an r of 10, not 20.

Let r₁ be the radius of the outer circle (diameter of 2 r₁) and r₂ be the radius of the inner circle. The areas are
A₁ = \(\displaystyle \pi\) r₁²
and
A₂ = \(\displaystyle \pi\) r₂²
A₂ is one-half A₁ so
\(\displaystyle \pi\) r₂² = \(\displaystyle \frac{1}{2}\pi\) r₁² = \(\displaystyle \frac{2}{4}\pi\) r₁²
or
\(\displaystyle r_2\, =\, \frac{\sqrt{2}}{2}\, r_1\) ~ 0.70710678118654 r₁
Since the diameters are just twice the radius, the diameters are the same ratio.

 

[gareth]

Hi Ishuda

From the first line of my question "A circle, say 20cm in diameter"
r=diameter/2=10
Area=pi*r²= pi*10²=314,1592cm²

btw using a "," instead of a "." is normal in numbers where I now live, these numbers were pasted from excel.
sorry for the confusion
Thank you

 

[Ishuda]

Hi Ishuda

From the first line of my question "A circle, say 20cm in diameter"
r=diameter/2=10
Area=pi*r²= pi*10²=314,1592cm²
why is the ratio = sin 45?
sorry for the confusion

Just means I should read the problem more carefully.