Given 3 circles with radius #r# positioned find the the radius #R# of the circumscribing circle in terms of the radius of the small circles #r#? If r = 5 cm what is the area of the circumscribing circle?

Question

Maverick Smith · Accepted Answer

Radius of each of three circles touching each other is r.

The side of the equilateral triangle formed by joining the three centers of the given three circles is #=2r#

Height of this equlateral triangle is # =sqrt((2r)^2-r^2)=sqrt3r#

Radius of the circumcircle of this equilateral triangle is #=2/3xxsqrt3r=(2sqrt3r)/3#

Hence Radius of the circle cicumscribing three circles will be #R= (2sqrt3r)/3+r=((2sqrt3)/3+1)r#

Area of this circle

#=piR^2=pi( (2sqrt3)/3+1)^2r^2#

#=(4/3+1+(4sqrt3)/3)pir^2#

#=1/3(7+4sqrt3)xx3.14xx5^2cm^2#

#=364.45cm^2#

Evelyn Arboleda · Answer

undefined The radius $ R $ of the circumscribing circle in terms of the radius of the smaller circles $ r $ is given by:

$$ R = \frac{r}{\sqrt{3}} $$

If $ r = 5 $ cm, then:

$$ R = \frac{5}{\sqrt{3}} $$

$$ R \approx 2.89 \, \text{cm} $$

The area $ A $ of the circumscribing circle can be found using the formula:

$$ A = \pi R^2 $$

Substitute the value of $ R $:

$$ A = \pi \times (2.89)^2 $$

$$ A \approx \pi \times 8.3521 $$

$$ A \approx 26.23 \, \text{cm}^2 $$