Question

Боковая грань правильной четырёхугольной пирамиды наклонена к плоскости основания под углом 60 градусов площадь основания пирамиды 16 найти боковую поверхность пирамиды.

Answer 1

Сторона основания = кор из 16 = 4
Радиус вписанной окружности (проекция апофемы) r = 2
Из оределения косинуса найдем апофему (апоф. )
r/апоф. =cos60
Апофема =2
(или r лежит протв угла 30 гр апоф = 1*2 = 2)
Sбок. = р*апоф. (р - полупериметр основания) = 8*2=16

Answer 2

Given Information

We are given the following information about a regular quadrilateral pyramid: - The lateral face of the pyramid is inclined to the base plane at an angle of 60 degrees. - The area of the base of the pyramid is 16.

Finding the lateral surface area of the pyramid

To find the lateral surface area of the pyramid, we need to know the slant height of the lateral face. Unfortunately, the given information does not directly provide the slant height. However, we can use the given angle and the side length of the base to find the slant height.

Let's denote the side length of the base as s and the slant height as l.

Using trigonometry, we can relate the side length of the base, the slant height, and the angle between the lateral face and the base. The formula is as follows:

l = s * sin(angle)

Now, we can substitute the given values into the formula to find the slant height:

l = s * sin(60 degrees)

To find the lateral surface area, we need to calculate the area of each lateral face and then sum them up. The lateral surface area of each face is given by:

A_lateral_face = (1/2) * s * l

Substituting the value of the slant height, we get:

A_lateral_face = (1/2) * s * (s * sin(60 degrees))

Since the pyramid is regular, all the lateral faces are congruent, so we can multiply the area of one face by the number of faces to get the total lateral surface area.

The number of lateral faces in a quadrilateral pyramid is 4.

Therefore, the lateral surface area of the pyramid is:

A_lateral_surface = 4 * A_lateral_face

Now, let's calculate the lateral surface area using the given information.

Calculation

Given: - Angle between the lateral face and the base = 60 degrees - Area of the base = 16

Using the formula for the slant height: l = s * sin(60 degrees)

Using the formula for the lateral surface area of each face: A_lateral_face = (1/2) * s * (s * sin(60 degrees))

The lateral surface area of the pyramid is: A_lateral_surface = 4 * A_lateral_face

Let's substitute the values and calculate the lateral surface area.

Solution

Using the given information, we can calculate the lateral surface area of the pyramid.

The side length of the base (s) can be found by taking the square root of the area of the base (16): s = sqrt(16) = 4

The slant height (l) can be calculated using the formula: l = s * sin(60 degrees) = 4 * sin(60 degrees)

Now, we can calculate the lateral surface area of each face: A_lateral_face = (1/2) * s * (s * sin(60 degrees)) = (1/2) * 4 * (4 * sin(60 degrees))

Finally, we can calculate the total lateral surface area of the pyramid: A_lateral_surface = 4 * A_lateral_face = 4 * [(1/2) * 4 * (4 * sin(60 degrees))]

Let's calculate the values:

s = 4 l = 4 * sin(60 degrees) A_lateral_face = (1/2) * 4 * (4 * sin(60 degrees)) A_lateral_surface = 4 * [(1/2) * 4 * (4 * sin(60 degrees))]

Using a calculator, we find: s = 4 l ≈ 6.9282 A_lateral_face ≈ 27.7128 A_lateral_surface ≈ 110.8512

Therefore, the lateral surface area of the pyramid is approximately 110.8512 square units.

Answer

The lateral surface area of the given quadrilateral pyramid is approximately 110.8512 square units.