Relationship Between the Side of an Equilateral Triangle and the Radius of Its Circumscribed Circle
The side length of an equilateral triangle equals the radius of its circumscribed circle multiplied by the square root of three. $$ l = r \cdot \sqrt{3} $$
A Practical Example
Let’s take the equilateral triangle ABC and the circle circumscribed around it, with center O and radius r.
In this case, the radius of the circumscribed circle is r = 2.3094.
We can apply the formula to find the length of a side of the equilateral triangle:
$$ l = r \cdot \sqrt{3} $$
$$ l = 2.3094 \cdot \sqrt{3} = 4 $$
This result gives us the side length of the equilateral triangle.
The Proof
To prove this relationship, let’s consider a regular hexagon inscribed in a circle with center O and radius r.
We choose three non-adjacent vertices of the hexagon - for example, A, E, and C - and connect them to form the central triangle AEC, along with three peripheral triangles (ABC, AEF, and CDE).
The peripheral triangles are congruent by the first triangle congruence theorem, since their sides match the lengths of the hexagon’s edges and the angle between them is 120°.
Therefore, all their sides are equal in length.
Note: A regular hexagon has all sides of equal length and all interior angles measuring 120°.
As a result, the central triangle AEC is an equilateral triangle.
Next, we want to calculate the length of any side of the equilateral triangle.
To do this, we draw segment CF, which is equal to the diameter of the circle. Thus, d = 2r.
Observing the figure, we see another triangle, ECF, which shares side EC with the equilateral triangle AEC.
The angle at vertex E of triangle ECF is a right angle because it’s an angle inscribed in a semicircle.
This means triangle ECF is a right triangle, allowing us to apply the Pythagorean theorem to determine the length of side EC:
$$ \overline{EC} = \sqrt{ \overline{FC}^2 - \overline{EF}^2 } $$
Here, segment FC represents the diameter of the circle, so d = 2r:
$$ \overline{EC} = \sqrt{ (2r)^2 - \overline{EF}^2 } $$
Segment EF corresponds to one side of the regular hexagon.
Since each side of a regular hexagon equals the radius of its circumscribed circle, we substitute EF = r:
$$ \overline{EC} = \sqrt{ (2r)^2 - r^2 } $$
Let’s simplify the expression step by step:
$$ \overline{EC} = \sqrt{ 4r^2 - r^2 } $$
$$ \overline{EC} = \sqrt{ 3r^2 } $$
$$ \overline{EC} = r \sqrt{ 3 } $$
Segment EC is one side of the equilateral triangle AEC and is therefore equal in length to the triangle’s other sides.
This demonstrates that the side of an equilateral triangle is equal to the radius of its circumscribed circle multiplied by the square root of three.
And so on.
