Circle section

Equilateral triangle with side 33 is inscribed circle section whose center is in one of the vertices of the triangle and the arc touches the opposite side.

Calculate:

a) the length of the arc
b) the ratio betewwn the circumference to the circle sector and the perimeter of the triangle

Correct result:

x =  29.9277 cm
y =  0.3023

Solution:

$$\begin{gathered} a=33\text{ cm } \\ r={\displaystyle \frac{\sqrt{3}}{2}}\cdot a={\displaystyle \frac{\sqrt{3}}{2}}\cdot 33\doteq 28.5788\text{ cm } \\ A=60^{\circ } \\ x=r\cdot A^{\circ }\to \text{ rad}=r\cdot A^{\circ }\cdot {\displaystyle \frac{\pi }{180}}=28.5788383249\cdot 60^{\circ }\cdot {\displaystyle \frac{3.1415926}{180}}=29.928=29.9277\text{ cm} \end{gathered}$$

$$\begin{gathered} p=3\cdot a=3\cdot 33=99\text{ cm } \\ y={\displaystyle \frac{x}{p}}={\displaystyle \frac{29.9277}{99}}=0.3023 \end{gathered}$$

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