A circle is circumscribed about an equilateral triangle with side lengths of 9 units each. What is the area of the circle, in square units? Express your answer in terms of pi.
27
To find the area of the circle, we need to find the radius of the circle first.
In an equilateral triangle, the radius of the circumscribed circle is equal to the length of a side divided by the square root of 3.
So, the radius of the circle is 9 / √3.
To find the area of the circle, we use the formula A = πr^2, where A is the area and r is the radius.
Substituting the value of the radius, we have A = π(9 / √3)^2.
Simplifying, we have A = π(81 / 3).
Further simplifying, we have A = 27π.
Therefore, the area of the circle is 27π square units.
To find the area of the circle, we first need to find the radius of the circle.
In an equilateral triangle, the radius of the circumscribed circle can be found by dividing the length of any side of the triangle by the square root of 3.
In this case, the side length of the equilateral triangle is 9 units. So, the radius of the circle can be found using the formula:
Radius = (Side length of the equilateral triangle) / (√3)
= 9 / (√3)
To simplify the expression, we can multiply both the numerator and denominator by (√3):
Radius = (9 * √3) / (√3 * √3)
= (9 * √3) / 3
= 3 * √3
Now that we have found the radius of the circle, we can find its area using the formula:
Area of a circle = π * (Radius)^2
= π * (3 * √3)^2
= π * (9 * 3)
= π * 27
Therefore, the area of the circle is 27π square units.
in an equilateral triangle, the altitudes are the medians.
So, the radius of the circle is 2/3 of the length of each altitude.
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