Find the perimeter and area of an equilateral triangle with a height of 24in.
Using the 30-60-90 triangle ratios
x/24 = 2/√3
x = 48/√3
perimeter = 2(48/√3) = 96/√3
area = (1/2)(48/√3)(24)
= 576/√3
To find the perimeter and area of an equilateral triangle, we need to know the length of one of its sides. However, you have given the height of the triangle, which is not sufficient to calculate these values directly.
In an equilateral triangle, the height bisects the base, creating two right-angled triangles. We can use the Pythagorean theorem to find the length of the base.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's label the sides of the right-angled triangle:
- The height of the triangle is 24 inches.
- One of the legs (adjacent to the right angle) is half of the base, which we'll call 'x.'
Using the Pythagorean theorem, we have:
x^2 + 24^2 = base^2
Now, let's solve for the base (or one side of the equilateral triangle):
x^2 + 576 = base^2.
Since it is an equilateral triangle, all sides are equal, so the base is equal to any side length 's'. Thus we now have:
x^2 + 576 = s^2.
To find the perimeter of an equilateral triangle, we know that all three sides are equal. Therefore, the perimeter is given by:
P = 3s.
To calculate the area of an equilateral triangle, we'll use the formula:
A = (sqrt(3)/4) * s^2.
Let's plug in the values we have:
x^2 + 576 = s^2
To calculate the side length 's,' we need to solve for 'x.' Here's how:
x^2 + 576 = s^2
x^2 = s^2 - 576
x = sqrt(s^2 - 576)
Once we find 'x,' we can substitute it back into the perimeter and area formulas:
P = 3s
P = 3(sqrt(s^2 - 576))
A = (sqrt(3)/4) * s^2
A = (sqrt(3)/4) * (s^2 - 576)
By substituting 's' with sqrt(s^2 - 576), you can get the perimeter and area of the equilateral triangle with a height of 24 inches.