Question

Сторона основания правильной треугольной пирамиды равна 6, а боковое ребро образует с плоскостью основания угол 45 градусов. Найти объем пирамиды.

Answer 1

Объем пирамиды

$V=\frac{1}{3}S_{ABC}H$

 

Т.к. ∆АВС - правильный, то его площадь

$S_{ABC}=\frac{AB^2\sqrt3}{4}=\frac{6^2\sqrt3}{4}=9\sqrt3$

 

Основание О высоты МО правильной пирамиды - центр описанной и вписанной окружностей. В правильном  ∆СКВ СК - медиана, биссекториса, высота.

В  ∆СКВ  КВ=3, ∠СВК=60°, СК = СВsin60°=6·√3/2=3√3

 

О - точка пересечения медиан ∆АВС, СО=2СК/3=2√3

 

Высота МО⊥(АВС), тогда МО⊥CO. ∆МОС - прямоугольный и ∠МСО=45°, значит и ∠СМО=45°. Значит, ∆МОС - равнобедренный, тогда СО=МО =2√3.

 

Таким образом, 

$V=\frac{1}{3}S_{ABC}H=\frac{1}{3}*9\sqrt3*2\sqrt3=18$

Answer 2

Given Information

We are given the following information about a right triangular pyramid: - The base side length is 6. - The angle between one of the lateral edges and the base plane is 45 degrees.

Calculating the Volume of the Pyramid

To find the volume of the pyramid, we need to know the height of the pyramid. Unfortunately, the given information does not directly provide the height. However, we can use the given angle and the base side length to calculate the height.

Let's denote the height of the pyramid as h and the base side length as a.

From the given information, we know that the angle between one of the lateral edges and the base plane is 45 degrees. This means that the lateral face of the pyramid is an isosceles right triangle.

Using trigonometry, we can find the height h of the pyramid. In an isosceles right triangle, the height is equal to half the length of the hypotenuse. In this case, the hypotenuse is the lateral edge of the pyramid.

Using the Pythagorean theorem, we can find the length of the hypotenuse (lateral edge) of the pyramid. Let's denote the hypotenuse as c.

From the Pythagorean theorem, we have:

c^2 = a^2 + a^2 = 2a^2

Taking the square root of both sides, we get:

c = sqrt(2a^2) = sqrt(2) * a

Since the height h is half the length of the hypotenuse, we have:

h = (1/2) * c = (1/2) * sqrt(2) * a

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

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

The base area of a right triangular pyramid is given by:

Base Area = (1/2) * base side length^2

Substituting the values, we have:

Volume = (1/3) * (1/2) * a^2 * [(1/2) * sqrt(2) * a]

Simplifying, we get:

Volume = (1/12) * sqrt(2) * a^3

Substituting the given value of the base side length (a = 6), we can calculate the volume of the pyramid.

Calculation

Using the given base side length (a = 6), we can calculate the volume of the pyramid as follows:

Volume = (1/12) * sqrt(2) * 6^3

Calculating the value, we get:

Volume ≈ 18.97 cubic units

Therefore, the volume of the pyramid is approximately 18.97 cubic units.

Answer

The volume of the pyramid is approximately 18.97 cubic units.