Formula Height Equilateral Triangle
2011
formula height equilateral triangle
What is the height of a pyramid that has an equilateral triangle for a base…?
…and 3 faces that are isosceles triangles with sides twice the length of the base? Lets say the base is 10 inches, making the facets 20 inches on edge. Please don’t answer if you know a formula or equation for this already.
In the first answer, the height of the pyramid for a 10 inch base was calculated to be greater than the length of the side. That is illogical. It must be shorter than 20 inches.
The second answer provided the height of the pyramid (y) as a factor of the base (x). The square root of 402 divided by six, or 3.34x is incorrect.
Suppose that the sides of the equilateral triangles are x and the sides of the isosceles triangle are 2x.
The height of the isosceles triangle is the slant height of the pyramid. Note that the area of each side of the pyramid is:
A = √[s(s - x)(s - 2x)(s - 2x)], by Heron’s Formula,
where s = (x + 2x + 2x)/2 = 5x/2.
Thus:
A = √[(5x/2)(5x/2 - x)(5x/2 - x)(5x/2 - 2x)] = (3√5/4)x^2.
Since the area of a triangle is (1/2)bh and the base of each isosceles triangle is x, we see that:
(1/2)xh = (3√5/4)x^2 ==> h = (3√5/2)x.
Now, to find the height of the pyramid, note that the height of the pyramid, the distance from the midpoint of one base to the center, and the slant height form a right triangle. Also, note that the medians of an equilateral triangle are also the altitudes of the triangles due to the SAS postulate.
The height of the equilateral triangle is (√3/2)x. Since the center of the triangle splits the median in a 1:2 ratio, we see that the distance from a side to the center is one-third of the height, or (√3/6)x. By the Pythagorean Theorem, if y is the height of the pyramid, then:
[(√3/6)x]^2 + y^2 = [(3√5/2)x]^2 ==> y = (√402/6)x.
Thus, the volume of the pyramid is:
V = (1/3)Bh, where B is the area of the equilateral base
= (1/3)[(√3/4)x^2][(√402/6)x]
= (√1206/72)x^3.
I hope this helps!
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