To find the area of an equilateral triangle when the height is given, we can use the formula for the area of a triangle:
Area = 1/2 × base × height
In an equilateral triangle, all sides are of equal length, and the height can be related to the base. However, first, we need to determine the length of the base using the height. The height (h) of an equilateral triangle relates to the side length (s) as follows:
h = (√3 / 2) × s
Given that the height (h) is 9 inches, we can rearrange the formula to solve for s:
s = (2/√3) × h
Substituting the height:
s = (2/√3) × 9 = 18/√3
Now that we have the side length, we can substitute the base into the area formula. Here, the base is equal to the side length (s):
Area = 1/2 × s × h
Now, substituting s and h into the area formula:
Area = 1/2 × (18/√3) × 9
Area = 81/√3
To express this in a simpler form, we can multiply the numerator and denominator by √3:
Area = (81√3) / 3 = 27√3
So, the area of the equilateral triangle with a height of 9 inches is approximately 46.76 square inches when you calculate 27√3, as √3 is roughly 1.732.