Question

основанием пирамиды служит треугольник,длины сторон которого равны 18,15 и 15.все боковые грани пирамиды образуют с ее основанием углы 45°.определить обьем этой пирамиды?помогите пожалуйста решить.........

Answer 1

Так как все грани пирамиды наклонены под одним углом, то вершина S проектируется в центр вписанной окружн. точка О, АК перпенд.ВС, ВК=КС=9

точка О принадл.АК, ОК радиус впис.окружности

AK найдем по т.Пифагора,АК=12,

S(АВС)=12*9=108

r(OK)=2S/p=2*108/(15+15+18)=9/2=4,5

SO=OK=9/2=4,5(треуг. SOK равнобедр.)

V=Sоснов.*SO/3

V=108*4,5/3=162

Answer 2

Calculation of the Volume of the Pyramid

To calculate the volume of the pyramid, we need to know the length of the base and the height of the pyramid. In this case, the base of the pyramid is a triangle with side lengths of 18, 15, and 15, and the angle between the base and the lateral faces is 45 degrees.

To find the height of the pyramid, we can use the formula for the height of a right triangle:

h = c * sin(A)

Where: - h is the height of the pyramid - c is the length of the side of the base triangle - A is the angle between the base and the lateral faces

In this case, the length of the side of the base triangle is 15, and the angle A is 45 degrees.

Let's calculate the height of the pyramid using the formula:

h = 15 * sin(45°)

Using a calculator, we find that the sine of 45 degrees is approximately 0.7071. Therefore, the height of the pyramid is:

h ≈ 15 * 0.7071 ≈ 10.6065

Now that we have the height of the pyramid, we can calculate its volume using the formula:

Volume = (1/3) * base area * height

The base area of the pyramid is the area of the triangle, which can be calculated using Heron's formula:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

Where: - s is the semi-perimeter of the triangle, calculated as (a + b + c) / 2 - a, b, and c are the side lengths of the triangle

In this case, the side lengths of the triangle are 18, 15, and 15. Let's calculate the semi-perimeter and the area of the triangle:

s = (18 + 15 + 15) / 2 = 24

Area = sqrt(24 * (24 - 18) * (24 - 15) * (24 - 15)) ≈ 108

Now, we can calculate the volume of the pyramid:

Volume = (1/3) * 108 * 10.6065 ≈ 359.55

Therefore, the volume of the pyramid is approximately 359.55 cubic units.

Please note that the calculations provided are based on the information given and the formulas used.

Answer 3

Calculation of the Volume of the Pyramid

To calculate the volume of the pyramid, we need to know the length of the base and the height. In this case, the base of the pyramid is a triangle with side lengths of 18, 15, and 15. The angles between the base and the lateral faces are 45 degrees.

To find the height of the pyramid, we can use the formula:

height = side * sin(angle)

where the side is the length of one of the sides of the base triangle and the angle is the angle between the base and the lateral face.

In this case, the side length is 15 and the angle is 45 degrees. Plugging these values into the formula, we get:

height = 15 * sin(45°)

To calculate the volume of the pyramid, we can use the formula:

volume = (1/3) * base_area * height

where the base_area is the area of the base triangle.

To find the area of the base triangle, we can use Heron's formula:

base_area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semiperimeter of the triangle and a, b, and c are the side lengths of the triangle.

In this case, the side lengths of the base triangle are 18, 15, and 15. Plugging these values into the formula, we get:

s = (18 + 15 + 15) / 2 = 24

base_area = sqrt(24 * (24 - 18) * (24 - 15) * (24 - 15))

Now we can calculate the height and the volume of the pyramid.

Calculation Steps:

1. Calculate the height of the pyramid: - height = 15 * sin(45°) 2. Calculate the area of the base triangle: - s = (18 + 15 + 15) / 2 = 24 - base_area = sqrt(24 * (24 - 18) * (24 - 15) * (24 - 15)) 3. Calculate the volume of the pyramid: - volume = (1/3) * base_area * height

Now let's calculate the height, base area, and volume of the pyramid.

Calculation:

1. Calculate the height: - height = 15 * sin(45°) = 15 * 0.7071 ≈ 10.6065

2. Calculate the base area: - base_area = sqrt(24 * (24 - 18) * (24 - 15) * (24 - 15)) ≈ 126.4911

3. Calculate the volume: - volume = (1/3) * base_area * height ≈ (1/3) * 126.4911 * 10.6065 ≈ 423.3037

Therefore, the volume of the pyramid is approximately 423.3037 cubic units.

Please note that the calculations provided are based on the given information and assumptions.