Answer:
radius = 3.31 in (3 significant figures)
Step-by-step explanation:
Let [tex]x[/tex] be one side of an equilateral triangle.
As an equilateral triangle has 3 sides of equal length,
Perimeter of equilateral triangle = [tex]x + x + x = 3x[/tex]
The circumscribed circle is the circle that passes through all three vertices of the triangle.
The altitude of a triangle refers to the line segment that can join the vertex of a triangle and the opposite side of the triangle in a way so that the line segment can be perpendicular to the opposite side of the triangle.
If we draw altitudes for all 3 vertices of the equilateral triangle, they will intersect in the center of the triangle, which is also the center of the circle (see attached diagram).
To calculate the radius in terms of [tex]x[/tex], we can use the right-angled triangle (highlighted in red on the attached diagram) and the trig ratio of [tex]cos(\theta)=\frac{A}{H}[/tex] where the angle = 30°, the adjacent side = [tex]\frac{1}{2} x[/tex] and the hypotenuse is the radius.
Therefore, [tex]cos(30)=\frac{0.5x}{r}[/tex] ⇒ [tex]r=\frac{0.5x}{cos(30)} =\frac{\sqrt{3}}{3} x[/tex]
Now we have an expression for the radius [tex]r[/tex], we can write an expression for the area of the circle:
Area of a circle = [tex]\pi r^2[/tex] = [tex]\pi (\frac{\sqrt{3}}{3} x)^2[/tex] = [tex]\pi \frac{1}{3} x^2[/tex]
We are told that the perimeter of the triangle equals the area of the circle, so using the 2 expressions we have determined:
perimeter of triangle = area of circle
[tex]6x=\pi \frac{1}{3} x^2[/tex]
Divide both sides by [tex]x[/tex]: [tex]6=\pi \frac{1}{3} x[/tex]
Multiple both sides by 3: [tex]18=\pi x[/tex]
Divide both sides by [tex]\pi[/tex]: [tex]x=\frac{18}{\pi }[/tex]
Therefore, to find the radius, substitute the found value of [tex]x[/tex] into the expression we found for the radius, [tex]r=\frac{\sqrt{3}}{3} x[/tex]
So the radius = [tex]\frac{\sqrt{3}}{3} \times\frac{18}{\pi }[/tex] = 3.307973373... = 3.31 in (3 significant figures)
