Question

find the area of a circle circumscribed about a equalateral triangle whose side is 18 inches long

Answer 1

There are different ways to solve for the circumscribed radius, depending on which part of geometry you are working on.

The most basic calculation taking advantage that the triangle ABC is equilateral is to construct a perpendicular bisector on side BC such that ABD is a right triangle right-angled at D.

AD is then the median, and the radius of the circumscribed circle is 2/3 of AD.

Using Pythagoras Theorem,
AD=sqrt(AB^2-BD^2)=sqrt(18^2-9^2)=sqrt(243)=9sqrt(3).

The radius of the circumscribed circle is therefore (2/3)*9sqrt(3)=6sqrt(3).


A more general method is
radius of circumscribed circle
=a/(2sin(A))
where a is any side of the triangle, and
A is the angle opposite side a.

Applied to the given equilateral triangle,
a=18
A=60°
so
r=18/(2sin(60)=6sqrt(3)