in order for the straight lines to be 90 deg with the circles, they must be radii of circles with same central point
the length of an arc is defined as c = r * θ (where r is the radius, and θ is the angle)
we define the inner circle with radius r₁ and its arc L₁ = r₁ * θ₁
we define the outer circle with radius r₂ and its arc L₂ = r₂ * θ₂
Because of (1), θ₁ + θ₂ = 2π
To create the shape, L₁ = L₂ = r₂ - r₁
If you start replacing and solving, you will get a 2nd grade quadratic, which has a positive and a negative solution. The positive solution is that magic number.
Take this shape as an example. The “square” in question consists of AC, BD, the outer AB, and the inner CD.
Point (5) means that, since the lines AC and BD are radii of the concentric circles, the arcs AB and CD should have the same inner angle. That’s because the angle COD is equal to AOB.
Since, the inner angle is the same, then the outer AOB should, by definition, be 2π - (the inner AOB), because that’s how radiants work; a circle is 2π rads.
Thank you, I need an evening with wikipedia to either understand what you wrote or ask more questions, because I don’t see why is that, but now I need to learn/remember that stuff
Solution:
Explanation:
in order for the straight lines to be 90 deg with the circles, they must be radii of circles with same central point
the length of an arc is defined as c = r * θ (where r is the radius, and θ is the angle)
we define the inner circle with radius r₁ and its arc L₁ = r₁ * θ₁
we define the outer circle with radius r₂ and its arc L₂ = r₂ * θ₂
Because of (1), θ₁ + θ₂ = 2π
To create the shape, L₁ = L₂ = r₂ - r₁
If you start replacing and solving, you will get a 2nd grade quadratic, which has a positive and a negative solution. The positive solution is that magic number.
Why point (5)?
Since the straight lines are radii, they cut the circles at angle θ and 2π - θ, respectively. Adding those, you get 2π.
Okay, but… Why? Is that a theorem that I don’t remember from school?
Take this shape as an example. The “square” in question consists of AC, BD, the outer AB, and the inner CD.
Point (5) means that, since the lines AC and BD are radii of the concentric circles, the arcs AB and CD should have the same inner angle. That’s because the angle COD is equal to AOB.
Since, the inner angle is the same, then the outer AOB should, by definition, be 2π - (the inner AOB), because that’s how radiants work; a circle is 2π rads.
Thank you, I need an evening with wikipedia to either understand what you wrote or ask more questions, because I don’t see why is that, but now I need to learn/remember that stuff
Thank you! But why arc CD and arc AB length should add to 2 PI? Or why does the angle COD times two is 2PI if that’s what you meant?
Point (5) is not about the arcs’ lengths. It’s about the angle they create with the center.
Also, I never said that COD * 2 = 2π. I said (inner COD) + (outer COD) = 2π rads