Find the area of triangle

Question

Find the area of triangle formed by joining centres of 3 circles of radius 6cm,14cm,15cm

Answer 1

Using Christina Y.'s solution, let a = 20, b = 29, c = 21, and                 s = ½(a+b+c) = 35.

 

Then, according to Heron's Formula, the area of the triangle is

 

  Δ = √[s(s-a)(s-b)(s-c)] = √[(35)(15)(6)(14)] = √44,100

 

                                                               = √441√100 = (21)(10)

 

                                                                                  = 210 

Answer 2

Hi Aarthi,

 

To answer this question, you have to assume that the three circles are touching each other. 

(It may help if you draw yourself a diagram).

 

The length of edges created by connecting the centers of these three circles equal the sum of the radius of adjacent circles. Therefore, the length of the three edges will be:

 

1: 6+14 = 20 cm

2: 14+15 = 29 cm

3: 6+15 = 21 cm

 

As you probably know, the formula for calculating the area of a triangle is:

 

(1/2)(length * height) = Area of the Triangle

 

At this point, you need the measurement of the height to calculate the area of the traingle.

To figure out the height of the triangle, you can use the Pythagorean theorem: a^2 + b^2 = c^2

Draw a line from the point where two of the circles meet, and extend the line to the tip of the triangle on the opposite side of that edge from which you extended the line from to create a 90 degrees. 

 

You know the length of the hypotenuse (which is c in P's theorem) and the length of one of the edges of the triangle, which you can use to calculate the height of the greater triangle. Use this height to calculate the area of the triangle created by joining the three centers :)

 

Hope this helps! Let me know if you need more help understanding how to solve this question.

 

----------------------------

 

Actually, I stand corrected by Mark. You cannot assume that the triangle is formed by 2 right triangles by extending the line.

 

You do have to use the Heron's formula in this case using the formula he has written below. 

 

Thus, knowing the measurement of your edges, you can get you s and subsitute the rest of your measurements into the formula. It is a two step process:

 

1. Calculate your S value by using this formula: s = 0.5 (a+b+c)

2. Calculate the area using the s value: Area = √( s(s-a)(s-b)(s-c) )

 

Thus, the area of the triangle should be:

 

s= 0.5(20 +29+ 21) = 35

A = √[35(35-20)(35-29)(35-21)] = 210 cm^2