Inner Curve Arc Length of a Window
The widows in one the oldest building in the university have a curved top. I noticed that the window frame has an equal width throughout, including around the arc at the top. My curiosity was to know which formula was used to cut the glass that fits into the window’s curved position.
I noticed that this is not a normal window because the top is not a semi-circle. First, I assumed that the arc length is part of a full circle. Let the length of the total frame labelled CD be 4 units while the width of the edges of the frame be 0.3 units. I choose a point P to be the horizontal centre on the frame. I then chose point P and labelled it as P(2,1) and then drew 2 concentric circles that were 0.3 units apart. The inner one was 4.7 units while the outer one was (4.7 + 0.3) 5 units. Remember that it is possible to choose any radii to represent the concentric circles. The only thing that should matter is that the outer and inner radii are 0.3 units apart.
The fundamental problem is to find the angle θ, then finding the arc length HI will be much easier.
Using the right triangle PMI, I computed angle α.
cos α =1.74.7 = 0.3617
The using the inverse ratio
α = arccos (0.3617) = 1.2007 radians.
The equivalent degree for this radians is = 68.79°
Because they lie in a straight line, 2α + θ = π (180°).
So angle θ = π – 2 × 1.2007 = 0.7402
Then the arc length HI can be calculated by applying the formula for arc length
s = r θ
s = 4.7 × 0.7402
= 3.4789
= 3.48 (2.d.p)
