Question

Помогитев основании треугольной пирамиды лежит прямоугольный треугольник с гипотенузой 2 и острым углом 30 градусов. Боковые ребра наклонены к плоскости основания под углом 60 градусов. Найдите площадь боковой поверхности пирамиды

Answer 1

Т.к. боковые рёбра наклонены  под одним углом, то эти рёбра будут равны, следовательно, равны и их проекции, т.е. основание высоты  равноудалено от вершин основания пирамиды, следовательно, это центр описанной окружности. Центром описанной окружности является середина гипотенузы т.е. проекции равны 30 см. Есть такое свойство: катет лежащий против угла в 30град. равен половине гипотенузы, т.е. наша гипотенуза - 60 см. Тогда высоту найдём как катет прямоугольного треугольника с другим катетом 30 см и противолежащим углом 30град. Н=30·tg30, H=30·1/√3=10√3 см 

Answer 2

Calculation of the lateral surface area of the pyramid

To find the lateral surface area of the pyramid, we need to calculate the area of each triangular face and then sum them up.

Given information: - The base of the pyramid is a right triangle with a hypotenuse of 2 and an acute angle of 30 degrees. - The lateral edges of the pyramid are inclined to the base plane at an angle of 60 degrees.

Let's break down the solution step by step:

1. Calculate the lengths of the sides of the right triangle: - The hypotenuse is given as 2. - The acute angle is 30 degrees. - We can use trigonometric functions to find the lengths of the other two sides of the right triangle.

Using the sine function, we can find the length of the side opposite the acute angle: ``` sin(30°) = opposite / hypotenuse ``` Rearranging the equation, we get: ``` opposite = sin(30°) * hypotenuse ``` Substituting the given values: ``` opposite = sin(30°) * 2 ```

Therefore, the length of the side opposite the acute angle is 2 * sin(30°).

2. Calculate the area of each triangular face: - The base of each triangular face is the side opposite the acute angle, which we calculated as 2 * sin(30°). - The height of each triangular face is the length of the lateral edge, which is given as 2.

The area of a triangle can be calculated using the formula: ``` Area = (base * height) / 2 ```

Substituting the values, we get: ``` Area = (2 * sin(30°) * 2) / 2 ```

Simplifying the equation, we have: ``` Area = 2 * sin(30°) ```

3. Calculate the lateral surface area of the pyramid: - Since the pyramid has four triangular faces, we need to multiply the area of one triangular face by 4 to get the total lateral surface area.

Therefore, the lateral surface area of the pyramid is: ``` Lateral Surface Area = 4 * (2 * sin(30°)) ```

Now, let's calculate the value of the lateral surface area using the given information:

Lateral Surface Area = 4 * (2 * sin(30°))

Using a calculator, we can find the value of sin(30°) as 0.5:

Lateral Surface Area = 4 * (2 * 0.5)

Simplifying the equation, we get:

Lateral Surface Area = 4 * 1

Therefore, the lateral surface area of the pyramid is 4 square units.

Please let me know if you need any further assistance!