Externally Tangent Circles Problem Solving

Today, we will solve problems involving externally tangent circles. If how to get their radii, their areas, the area they covered and more. We will demonstrate techniques how to solve them easily. Don’t forget to like us on Facebook or follow us on twitter to get updates. You might also want to try our current problem of the week.

externally tangent circles

Given the circles above, if the sum of the segments connecting their centers are 8,9,11 respectively.

  1. What are the radii of 3 circles
  2. What is the area of the shaded region?


According to one of the theorems on circle, the line connecting the centers of two tangent circles passes their point of tangency.

Thus, we can create the following relations.

r_1+r_2=8 (i)

r_2+r_3=9 (ii)

r_1+r_3=11 (iii)

We have three equations and just enough to solve for three unknown radii.

Adding three equations we have,

r_1+r_2+ r_2+r_3+r_1+r_3=8+9+11


r_1+r_2+r_3=14 (iv)

To solve for the radius just subtract the 1st,2nd, and 3rd equations from the fourth equation.

Solving for r_3: iv-i

r_1+r_2+r_3-( r_1+r_2)=14-8



Solving for r_1: iv-ii

r_1+r_2+r_3-( r_2+r_3)=14-9



Solving for r_2: iv-iii

r_1+r_2+r_3-( r_1+r_3)=14-11


Since we already have the radii of three circles, we will be able to solve the circumference and area of circles if asked.

Trick and Technique:

The next time you encounter the same problem. Add the three given segments, divide the sum by 2 and subtract the individual given segments from the quotient to get the individual radius of the circle.

In our problem above, 8+9+11=28, 28÷2=14. 14-8=6, 14-9=5, and 14-11=3.

To solve for the area of the shaded region, find the area of the triangle formed by connecting the radii of three circles using heron’s formula and subtract the sum of the areas of three sectors formed by three circles.

The formula for the area of sectors is A=\displaystyle\frac{r^2\theta}{2}. Where the angle \theta is in radians.

We already have the radii of the circles, to find the angle use cosine law and the rest is substitution.




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