# Find the area enclosed by the arc and string of a circle when the arc length is constant

The calculation method differs depending on the angle θ, and the calculation is divided into three types: 0 ° <θ <180 °, θ = 180 °, 180 ° <θ <360 °

Calculation method at 0 ° <θ <180 ° is
(Fan area in Fig.2)－(Triangle area in Fig.4)

Calculation method at θ ＝180 ° is simple,
Half the area of a circle

Calculation method at 180 ° <θ <360 ° is
Area of circle － {(Fan area of Fig.2) － (Triangle area of Fig.4)}

Here, I will organize the symbols of each size.
・Since the arc length is constant, the result is shown in Fig.3 .
By the way, the relationship between arc length, radius, and angle is that the angle is in radians [rad] instead of degrees [°, deg], so
arc length [mm] = radius [mm] x angle [rad]

・Fig.5 shows each side of the triangle
The relationship between the sides of the triangle and sin θ and cos θ is shown in Fig.2 of the article.

From the above, the area is expressed by the formula in Fig.6 .
When the condition of r1 = 1 is added to this formula and calculation is performed using Excel at 1 ° intervals, the graph in Fig.7 is obtained.

0°<θ<180° での計算方法は
（図２の扇形の面積）－　（図４の三角形の面積）

θ=180° の計算方法は
円の面積の半分

180°<θ<360° での計算方法は

ここで各寸法の記号を整理していく
・円弧長さは一定という条件なので、図３となる
ちなみに円弧長さと半径、角度の関係は、角度を度[°、deg]単位ではなくラジアン[rad]としているので　円弧長さ[mm]＝半径[mm]×角度[rad]　となる

・三角形の各辺は図５となる

この式にr1=1という条件を加えて、1°間隔でエクセルを使って計算すると図７のグラフとなる。