How do I find the angle between lateral edges of a regular triangular pyramid if the edge is 60m and the height is 18m? Please help!
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math.stackexchange.com/questions/2649250/compute-the-dihedral-angle-of-a-regular-pyramid
To find the angle between the lateral edges of a regular triangular pyramid, we can use trigonometry.
First, let's calculate the slant height (l) of the triangular pyramid. The slant height is the hypotenuse of a right triangle formed by the lateral edge and the height. We can use the Pythagorean theorem to find l.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the slant height (l), the base is half the edge length of the triangular pyramid, and the height is given.
The base of the right triangle is half the edge length because the base of the triangular pyramid is an equilateral triangle, and the slant height connects the top vertex (apex) with the midpoint of one of the sides.
So, the base of the right triangle is: (1/2) * 60m = 30m
Now, using the Pythagorean theorem, we have:
l^2 = 30^2 + 18^2
Simplifying this equation, we get:
l^2 = 900 + 324
l^2 = 1224
To find l, we can take the square root of both sides:
l = √1224
l ≈ 34.96m
Now that we have the slant height (l) and the height (h), we can use the trigonometric function cosine to find the angle between the lateral edges.
Cosine is defined as the adjacent side divided by the hypotenuse in a right triangle. In this case, the adjacent side is the height (h) and the hypotenuse is the slant height (l).
So, the equation for cosine is:
cos(angle) = h / l
Plugging in the values we have:
cos(angle) = 18 / 34.96
Now, we can find the angle by taking the inverse cosine (or arccos) of both sides:
angle = arccos(18 / 34.96)
Calculating this using a calculator, we find:
angle ≈ 56.26°
Therefore, the angle between the lateral edges of the regular triangular pyramid is approximately 56.26°.