Question

Боковое ребро правильной треугольной пирамиды равно 6 см и составляет с плоскостью основания угол в 60 градусов. Найти объем пирамиды.

Answer 1

находим высоту пирамиды 6*sin60=6*sqrt(3)/2=3*sqrt(3)

Находим площадь основания

S=3R^2sqrt(3)/4

R=6*cos60=3

S=3*9sqrt(3)/4=27sqrt(3)/4

V=1/3hS=27*sqrt(3)*3sqrt(3)/3*4=81/4=20,25

Answer 2

Given Information:

We are given that the length of one of the lateral edges of a regular triangular pyramid is 6 cm, and it forms a 60-degree angle with the base plane.

Calculating the Volume of the Pyramid:

To find the volume of the pyramid, we need to know the height of the pyramid. However, the given information does not directly provide the height.

To solve this, we can use trigonometry to find the height of the pyramid. Since the given angle is 60 degrees and the length of the lateral edge is 6 cm, we can use the sine function to find the height.

Let's denote the height of the pyramid as h. The sine of the angle is defined as the ratio of the opposite side (height) to the hypotenuse (lateral edge). Therefore, we have:

sin(60 degrees) = h / 6

Simplifying the equation, we get:

h = 6 * sin(60 degrees)

Using a calculator, we can find that sin(60 degrees) = 0.866. Substituting this value into the equation, we get:

h = 6 * 0.866 = 5.196 cm

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 of the pyramid is an equilateral triangle, and we know that the length of each side of the triangle is 6 cm. The formula for the area of an equilateral triangle is:

Area = (sqrt(3) / 4) * side^2

Substituting the given side length of 6 cm into the formula, we get:

Area = (sqrt(3) / 4) * 6^2 = 9sqrt(3) cm^2

Now we can calculate the volume of the pyramid:

Volume = (1/3) * 9sqrt(3) cm^2 * 5.196 cm

Evaluating this expression, we find:

Volume ≈ 28.84 cm^3

Therefore, the volume of the given pyramid is approximately 28.84 cubic centimeters.