Ratio of sides

Calculate the area of a circle that has the same circumference as the circumference of the rectangle inscribed with a circle with a radius of r 9 cm so that its sides are in ratio 2 to 7.

Result

S =  157.617 cm2

Solution:

$r=9 \ \text{cm} \ \\ r^2=(a/2)^2 + (b/2)^2 \ \\ a:b=2:7 \ \\ \ \\ r^2=(a/2)^2 + ((7/2 \cdot \ a)/2)^2 \ \\ r^2=a^2/4 + a^2 \cdot \ (7/4)^2 \ \\ \ \\ a=r / (\sqrt{ 1/4+(7/4)^2 })=9 / (\sqrt{ 1/4+(7/4)^2 }) \doteq 4.945 \ \text{cm} \ \\ \ \\ b=7/2 \cdot \ a=7/2 \cdot \ 4.945 \doteq 17.3074 \ \text{cm} \ \\ \ \\ o=2 \cdot \ (a+b)=2 \cdot \ (4.945+17.3074) \doteq 44.5048 \ \text{cm} \ \\ \ \\ o=2 \pi \cdot \ r_{1} \ \\ \ \\ r_{1}=o / (2 \pi)=44.5048 / (2 \cdot \ 3.1416) \doteq 7.0832 \ \text{cm} \ \\ \ \\ \ \\ S=\pi \cdot \ r_{1}^2=3.1416 \cdot \ 7.0832^2 \doteq 157.6174 \doteq 157.617 \ \text{cm}^2$

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