find the area of an equilateral triangle with an apothem having a length of 4 feet. round to the nearest tenth
let the half-base be x
tan60 = 4/x
x = 4/tan60 = 4/√3
area = (1/2)(2x)(4) = 4x = 16/√3 or appr 9.2 ft^2
To find the area of an equilateral triangle, you need to know the length of its side. The apothem, which is the distance from the center of the triangle to its side, is not enough to determine the area.
However, we can find the length of the side using the apothem.
In an equilateral triangle, the apothem bisects the side and forms a right angle with the side. This creates a right triangle where the apothem is the hypotenuse.
Using Pythagorean theorem, we can find the length of the side.
Let's call the length of the side "s" and the apothem "a". We know that the apothem is 4 feet.
Using Pythagorean theorem: s^2 = a^2 + (s/2)^2
Plugging in the values: s^2 = 4^2 + (s/2)^2
Expanding and simplifying: s^2 = 16 + (s^2/4)
Get rid of the fraction: 4s^2 = 64 + s^2
Combine like terms: 3s^2 = 64
Divide both sides by 3: s^2 = 64/3
Take the square root of both sides: s ≈ √(64/3)
Now we have the length of the side, which is approximately 5.89 feet.
To find the area of the equilateral triangle, we can use the formula: Area = (sqrt(3)/4) * s^2
Plugging in the values: Area ≈ (sqrt(3)/4) * (5.89)^2
Calculating: Area ≈ (1.732/4) * 34.6721
Rounding to the nearest tenth: Area ≈ 14.7 square feet
Therefore, the area of the equilateral triangle with an apothem of 4 feet is approximately 14.7 square feet.