Finding the Circumcircle Radius of an Equilateral Triangle

This problem asks us to find the radius of the circumcircle for a specific equilateral triangle. We are given the side length of the equilateral triangle as $\sqrt{3}$ units.

Understanding the Circumcircle and Circumradius

The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. Its center is called the circumcenter, and its radius is called the circumradius. In an equilateral triangle, the circumcenter coincides with the centroid, incenter, and orthocenter.

Formula for Circumradius of an Equilateral Triangle

For an equilateral triangle with side length 'a', the circumradius (R) is given by the formula:

$\qquad R = \frac{a}{\sqrt{3}}$

Alternatively, if 'h' is the altitude (height) of the equilateral triangle, the circumcenter divides the altitude in a 2:1 ratio, with the longer part being the circumradius. The altitude of an equilateral triangle is $h = \frac{\sqrt{3}}{2}a$. Thus, $R = \frac{2}{3}h = \frac{2}{3} \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{3}a = \frac{a}{\sqrt{3}}$.

Calculating the Circumradius

We are given the side length of the equilateral triangle, $a = \sqrt{3}$ units.

Using the formula $R = \frac{a}{\sqrt{3}}$, we substitute the given value of 'a':

$\qquad R = \frac{\sqrt{3}}{\sqrt{3}}$

Now, we simplify the expression:

$\qquad R = 1$

So, the radius of the circumcircle of the equilateral triangle with $\sqrt{3}$ unit side is 1 unit.

Step-by-Step Solution

1. Identify the given information: Side length of equilateral triangle $a = \sqrt{3}$ units.
2. Recall the formula for the circumradius (R) of an equilateral triangle: $R = \frac{a}{\sqrt{3}}$.
3. Substitute the given side length into the formula: $R = \frac{\sqrt{3}}{\sqrt{3}}$.
4. Calculate the result: $R = 1$.
5. State the final answer: The circumcircle radius is 1 unit.

Comparing our result with the given options, we find that 1 unit matches one of the choices.

Revision Table: Equilateral Triangle Properties

Additional Information about Equilateral Triangles and Circles

In an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are all the same point. This point is equidistant from all sides (inradius) and all vertices (circumradius).

The relationship between the circumradius (R) and the inradius (r) in an equilateral triangle is particularly simple: $R = 2r$. We can see this from the formulas:

• $R = \frac{a}{\sqrt{3}}$
• $r = \frac{a}{2\sqrt{3}}$

Dividing R by r, we get $\frac{R}{r} = \frac{a/\sqrt{3}}{a/(2\sqrt{3})} = \frac{a}{\sqrt{3}} \times \frac{2\sqrt{3}}{a} = 2$. So, $R = 2r$. The circumcenter is also the centroid, which divides each median (and altitude/angle bisector) in a 2:1 ratio. The circumradius is the larger part (2), and the inradius is the smaller part (1) of the altitude.