find the area of a circle circumscribed about a equalateral triangle whose side is 18 inches long
To find the area of a circle circumscribed about an equilateral triangle, we need to follow these steps:
1. Determine the radius of the circumscribed circle.
2. Calculate the area of the circle using the formula for the area of a circle.
Step 1: Determine the radius of the circumscribed circle.
In an equilateral triangle, the radius of the circumscribed circle is equal to one-third of the length of any side.
Given that the side of the equilateral triangle is 18 inches, the radius of the circumscribed circle can be calculated as:
radius = 1/3 * side
radius = 1/3 * 18 inches
radius = 6 inches
Step 2: Calculate the area of the circle using the formula for the area of a circle.
The formula for the area of a circle is:
Area = π * radius^2
Plugging in the calculated radius:
Area = π * (6 inches)^2
Area = π * 36 square inches
Therefore, the area of the circle circumscribed about the equilateral triangle with a side length of 18 inches is 36π square inches, or approximately 113.1 square inches.
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)