Question

Высота правильной четырехугольной пирамиды равна 4 м, а диагональ основания - 10 м. найти площадь ее боковой поверхности.

Answer 1

DC²+BC²=DB²
DC=BC
2DC²=100
DC=5√2
HD=(1/2)DB=5
в треугольнике SHD
SD²=SH²+HD²
SD=√41
DE=(1/2)DC=(5√2)/2
в треугольнике SDE
SE²=SD²-DE²
SE=(√114)/2
S(боковой поверхности)=4S(SDC)=4·SE·DC·(1/2)=10√57

Answer 2

Calculating the lateral surface area of a regular quadrilateral pyramid

To find the lateral surface area of a regular quadrilateral pyramid, we can use the formula:

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

Given: - Height of the pyramid (h) = 4 m - Diagonal of the base (d) = 10 m

First, we need to find the perimeter of the base and the slant height.

Finding the perimeter of the base

The base of a regular quadrilateral pyramid is a square. The diagonal of a square can be related to its side length using the formula: d = s * √2, where d is the diagonal and s is the side length.

From source we have the diagonal of the base (d) = 10 m. Using the formula, we can find the side length (s) of the base.

s = d / √2

s = 10 / √2

Calculating the side length of the base

Let's calculate the side length of the base:

s ≈ 7.07 m

Finding the perimeter of the base

The perimeter of a square is given by: Perimeter = 4 * s

Perimeter = 4 * 7.07

Calculating the perimeter of the base

Let's calculate the perimeter of the base:

Perimeter ≈ 28.28 m

Finding the slant height

The slant height (l) can be found using the Pythagorean theorem, where l is the slant height, s is the side length of the base, and h is the height of the pyramid.

The formula is: l = √(s^2 + h^2)

Calculating the slant height

Let's calculate the slant height:

l = √(7.07^2 + 4^2)

l ≈ √(49.9849 + 16)

l ≈ √65.9849

l ≈ 8.12 m

Calculating the lateral surface area

Now that we have the perimeter of the base and the slant height, we can use the formula to find the lateral surface area:

LSA = (28.28 * 8.12) / 2

Calculating the lateral surface area

Let's calculate the lateral surface area:

LSA ≈ 114.97 square meters

Conclusion

The lateral surface area of the regular quadrilateral pyramid is approximately 114.97 square meters.