For the square pyramid, the lateral edge length, e , is 25 in., and the slant height, s , is 24 in. Find the height of the pyramid, h . Round the answer to the nearest whole number. Hint: Before finding the length of h , you will first need to find the length of r The height of the square pyramid is?
To solve this problem, we can use the Pythagorean Theorem.
Let's consider one of the triangular faces of the pyramid. We can see that the side length of the base is the same as the lateral edge length e, which is 25 inches.
Let r be the height of one of the triangles (which is also the height of the pyramid).
Using the Pythagorean Theorem, we can find r:
r^2 + (e/2)^2 = s^2
r^2 + (25/2)^2 = 24^2
r^2 + 12.5^2 = 576
r^2 + 156.25 = 576
r^2 = 576 - 156.25
r^2 = 419.75
r ≈ √419.75
r ≈ 20.49
Now that we have the height of the triangle (r), we can use it to find the height of the pyramid. Since the height of the pyramid is the same as the height of the triangular faces, the height of the pyramid is approximately 20 inches.
To find the height of the square pyramid (h), we can use the Pythagorean theorem.
Given:
Lateral edge length (e) = 25 in.
Slant height (s) = 24 in.
Step 1: Find the length of the apothem (r).
The apothem (r) can be found using the Pythagorean theorem applied to one of the triangle faces.
r^2 = s^2 - (e/2)^2
r^2 = 24^2 - (25/2)^2
r^2 = 576 - (625/4)
r^2 = 576 - 390.625
r^2 = 185.375
r ≈ √185.375
r ≈ 13.61 in. (rounded to two decimal places)
Step 2: Find the height (h) using the apothem (r) and lateral edge length (e).
h = √(s^2 - r^2)
h = √(24^2 - 13.61^2)
h = √(576 - 185.375)
h = √390.625
h ≈ 19.77 in. (rounded to two decimal places)
Therefore, the height of the pyramid (h) is approximately 20 in. (rounded to the nearest whole number).
To find the height of the square pyramid, we first need to find the length of the slant height, r. From there, we can use the Pythagorean theorem to solve for the height, h.
Step 1: Finding the length of r
The slant height, s, is given as 24 inches. The lateral edge length, e, is given as 25 inches. To find the length of r, we can use the formula:
r = sqrt(e^2 - (s/2)^2)
Substituting the given values:
r = sqrt(25^2 - (24/2)^2)
r = sqrt(625 - 144)
r = sqrt(481)
r ≈ 21.93 inches
Step 2: Finding the height, h
Now that we have the length of r, we can use the Pythagorean theorem to solve for the height, h. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (r) is equal to the sum of the squares of the lengths of the other two sides.
Using the information given, we have a right triangle with sides r, h, and s/2. Applying the Pythagorean theorem, we get:
r^2 = h^2 + (s/2)^2
Substituting the known values:
(21.93)^2 = h^2 + (24/2)^2
481 ≈ h^2 + 144
h^2 ≈ 481 - 144
h^2 ≈ 337
h ≈ sqrt(337)
h ≈ 18.36 inches
Rounding to the nearest whole number, the height of the square pyramid is approximately 18 inches.