The Equilateral Triangle

 

Finding the radius, R, of the circumscribing circle is equivalent to finding the distance from the centroid of the triangle to one of the vertices.  Finding the radius, r, of the inscribed circle is equivalent to finding the distance from the centroid to the midpoint of one of the sides.

 

If each vertex is connected to the midpoint of the opposite side by a straight line, then the lines intersect at the centroid of the triangle.

 

From Figure 3.1, R is the distance from the centroid to a vertex and r is the distance from the centroid to the midpoint of a side. 

 

We have the following relationships:

 

 

 

From the table of trigonometric functions of common angles, we have

 

 

 

Therefore we have

 

 =

 

and so

 

3.1)                  =

 

where a is the length of a side and r is the radius of the inscribed circle.


 

 

For R we have

 

 = 

 

and so

 

3.2)                  = 

 

where a is the length of a side and R is the radius of the circumscribing circle.

 

 

In order to find a in terms of R we just turn Equation 3.2 around like this:

 

  so 

 

But    so  we have

 

 

So we finally have a in terms of R as

 

3.3)                 

 

where a is the length of a side and R is the radius of the circumscribing circle.

 

 

To find r in terms of R recall that from Equation 3.1 we have

 

 


Substituting the value of a just found in this equation gives us

 

 

And so we have

 

3.4)                 

 

where r is the radius of the inscribed circle and R is the radius of the circumscribed circle.

 

 


 

Figure 3.1

 

Equilateral triangle.  All sides are equal and all angles are 60 degrees.

 

 

 

 

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