What is the perimeter of an equilateral triangle with an altitude of 15m?
s = 2h/√3, so 3s = 2h√3 = 30√3 m
To find the perimeter of an equilateral triangle with an altitude of 15m, we need to determine the length of one side of the triangle.
First, let's understand the properties of an equilateral triangle. In an equilateral triangle, all three sides are equal in length, and all three angles are equal, measuring 60 degrees.
To calculate the length of one side, we can use the altitude. Since the altitude creates a right triangle, we can apply the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the altitude of the equilateral triangle forms a right triangle with one side being half the length of the base side of the triangle. Thus, the length of the base side is twice the length of the altitude.
Let's denote the length of one side of the equilateral triangle as "s". Therefore, the base side would be "2s".
According to the Pythagorean theorem, the equation for the right triangle formed by the altitude is:
(2s)^2 = s^2 + 15^2
Simplifying the equation:
4s^2 = s^2 + 225
3s^2 = 225
Dividing both sides by 3:
s^2 = 75
Taking the square root of both sides:
s = √75
Simplifying the square root:
s ≈ 8.66 (rounded to two decimal places)
Now that we have the length of one side, we can calculate the perimeter of the equilateral triangle by multiplying the length of one side by 3:
Perimeter = 3s
Substituting the value of "s":
Perimeter ≈ 3 * 8.66
Perimeter ≈ 25.98 meters
Therefore, the perimeter of an equilateral triangle with an altitude of 15m is approximately 25.98 meters.