# The entrance hall to the Louvre Museum

in Paris is a glass and metal pyramid with a surface area of approximately 1980 square on each side . The base of the pyramid is a square that is 35.4 meters on each side. What is the slant height of the pyramid ? Round to the nearest hundredth . The base area is not included in the surface area.

## let the slant height of one side be h

then using (1.2)base x height
1980 = (1/2)(35.4)h
h = 3960/(35.4)
= 111.8644

= appr 111.86 m

this can't be. According to the given dimensions on the webpage, the base is about 35 by 35 and the height is 21.6 m, for a slant height of 27.8 m
this would give each triangle an area of
(1/2)(35)(27.8)
= 486.5

for a total of 486.5(4) or 1946 square units
You stated an area of 1980 for each side.

area of one triangle = 1980/4 = 495
then (1/2)(35.4)h = 495
h = 27.966

height is 27.97 m

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## To find the slant height of the pyramid, we can use the Pythagorean theorem. Let's start by calculating the height of the pyramid.

Since the base of the pyramid is a square with a side length of 35.4 meters, its area is calculated as:

Base area = side length^2
Base area = 35.4^2
Base area = 1253.16 square meters

We are given that the total surface area of the pyramid (excluding the base area) is approximately 1980 square meters on each side. Since the base area is not included, the remaining surface area is only the four triangular sides of the pyramid.

Surface area of all four triangular sides = Total surface area - Base area
Surface area of all four triangular sides = 1980 - 1253.16
Surface area of all four triangular sides = 726.84 square meters

To calculate the height (h) of the triangle, we can use the formula for the area of a triangle which is:

Area of a triangle = 1/2 * base * height

Since the triangular sides of the pyramid are all equilateral triangles, the base of the triangle is equal to each side length of the square base. So, the base of the triangle is 35.4 meters.

Now we can calculate the height (h) of the triangle:

Area of a triangle = 1/2 * base * height
726.84 = 1/2 * 35.4 * h

Simplifying the equation:

726.84 = 17.7h

Dividing both sides of the equation by 17.7:

h = 726.84 / 17.7
h = 41.04

Therefore, the height of the triangular side of the pyramid is approximately 41.04 meters.

To find the slant height (l) of the pyramid, we can use the Pythagorean theorem. The slant height (l) is the hypotenuse of a right triangle with the height (h) as one side and half the diagonal of the square base as the other side.

The diagonal of the square base can be calculated using the Pythagorean theorem as well. The length of the diagonal (d) is equal to the side length of the square multiplied by the square root of 2 (√2).

Diagonal of the square base (d) = side length * √2
Diagonal of the square base (d) = 35.4 * √2
Diagonal of the square base (d) ≈ 50.10

Now we can calculate the slant height (l) using the Pythagorean theorem:

l^2 = h^2 + (d/2)^2
l^2 = 41.04^2 + (50.10/2)^2
l^2 = 1682.0416 + 1252.5025
l^2 ≈ 2934.5441

Taking the square root of both sides of the equation:

l ≈ √2934.5441
l ≈ 54.16

Therefore, the slant height of the pyramid is approximately 54.16 meters when rounded to the nearest hundredth.

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## To find the slant height of the pyramid, we can use the formula for the lateral surface area of a pyramid:

Lateral Surface Area = (1/2) * Perimeter of base * Slant height

In this case, the base of the pyramid is a square with a side length of 35.4 meters.

The perimeter of the base can be found by multiplying the side length by 4:

Perimeter of base = 35.4 meters * 4 = 141.6 meters

The given surface area of the pyramid is 1980 square meters on each side, which means the total lateral surface area can be found by multiplying it by 4:

Lateral Surface Area = 1980 square meters * 4 = 7920 square meters

Now, we can rearrange the formula and solve for the slant height:

Slant height = (Lateral Surface Area) / ((1/2) * Perimeter of base)

Slant height = 7920 square meters / ((1/2) * 141.6 meters)

Slant height = 7920 square meters / 70.8 meters

Slant height ≈ 111.86 meters (rounded to two decimal places)

Therefore, the slant height of the pyramid is approximately 111.86 meters.