Find the relation between a side and the circumradius of an equilateral triangle.

Hint: An equilateral triangle is a triangle that has all its sides equal in length. Also, the three angles of the equilateral triangle are congruent and equal to 60 degrees and circumradius is defined as the radius of that circle which surrounds the triangle and hence, we can find the relation between a side and circumradius by considering its interior angle.

Complete step by step solution:
As given we need to find the relation between a side and the circumradius of an equilateral triangle.
For a triangle,
 \[\dfrac{a}{{\operatorname{Sin} A}} = 2R\]
In which R is the circumradius.
Here, \[A = 60^\circ \] ;
Since it is an Equilateral triangle, its interior angle is equal to 60 degrees, hence we need to find sin 60 degrees i.e.,
 \[\dfrac{a}{{\sin A}} = 2R\]
 \[\dfrac{a}{{\sin \left( {{{60}^\circ }} \right)}} = 2R\]
We know that, \[\sin {60^\circ } = \dfrac{{\sqrt 3 }}{2}\]
 \[\dfrac{a}{{\dfrac{{\sqrt 3 }}{2}}} = 2R\]
 \[a = \dfrac{{\sqrt 3 }}{2}\left( 2 \right)R\]
 \[a = \sqrt 3 R\]
Therefore, the size of the equilateral triangle is \[\sqrt 3 \] of the circumradius.

Note: The key point to find an equilateral triangle is that we know that all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°, hence we need to consider its interior angle of the equilateral triangle to find the relation.