Find the area of a circle circumscribed about an equilateral triangle whose side is 18 inches long.
108pi
To find the area of a circle circumscribed about an equilateral triangle, we need to determine the radius of the circle.
In an equilateral triangle, all three sides are equal, and each angle measures 60 degrees. The radius of the circumscribed circle is the distance from the center of the circle to any vertex of the triangle.
To find the radius, we can use the properties of an equilateral triangle.
The height of an equilateral triangle can be found using the formula:
height = side * sqrt(3) / 2
Substituting the given value, the height of the equilateral triangle is:
height = 18 * sqrt(3) / 2
The radius of the circumscribed circle is equal to the height of the equilateral triangle. Therefore, the radius of the circle is:
radius = 18 * sqrt(3) / 2
To find the area of the circle, we can use the formula:
Area = π * radius^2
Substituting the value of the radius, the area of the circle is:
Area = π * (18 * sqrt(3) / 2)^2
Simplifying further:
Area = π * (9 * 3)
Area = 27π
Therefore, the area of the circle circumscribed about an equilateral triangle whose side is 18 inches long is 27π square inches.