Question

Высота конуса равна два с корня трех см , а радиус основания 4 см. Найти площадь полной поверхности правильной треугольной пирамиды , вписанной в конус. Нарисовать рисунок! Спасибо за помощь!

Answer 1

Calculation of the Surface Area of the Triangular Pyramid Inscribed in the Cone

To find the surface area of the triangular pyramid inscribed in the cone, we need to calculate the areas of the individual components and then sum them up.

Let's start by drawing a diagram to visualize the problem:

           /\
          /  \
         /    \
        /______\

In the diagram, the cone is represented by the larger triangle, and the triangular pyramid is represented by the smaller triangle inscribed within the cone.

Given information: - Height of the cone = 2 + √3 cm - Radius of the base of the cone = 4 cm

To find the surface area of the triangular pyramid, we need to calculate the areas of the base and the lateral faces.

Calculation of the Base Area

The base of the triangular pyramid is an equilateral triangle inscribed in the base of the cone. The side length of the equilateral triangle can be calculated using the radius of the cone's base.

The formula to calculate the side length of an equilateral triangle inscribed in a circle is:

Side length = 2 * (radius of the circle) * (sqrt(3))

In this case, the radius of the circle is equal to the radius of the cone's base, which is 4 cm.

Substituting the values into the formula, we get:

Side length = 2 * 4 cm * sqrt(3) = 8 cm * sqrt(3)

The area of an equilateral triangle can be calculated using the formula:

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

Substituting the value of the side length, we get:

Area = (sqrt(3) / 4) * (8 cm * sqrt(3))^2 = (sqrt(3) / 4) * (64 cm^2 * 3) = 16 cm^2 * sqrt(3)

Therefore, the area of the base of the triangular pyramid is 16 cm^2 * sqrt(3).

Calculation of the Lateral Surface Area

The lateral surface area of the triangular pyramid can be calculated by finding the sum of the areas of the three triangular faces.

Each triangular face is an isosceles triangle with two equal sides and an angle of 60 degrees between them. The height of each triangular face can be calculated using the height of the cone.

The formula to calculate the height of an isosceles triangle given the length of the equal sides and the angle between them is:

Height = (side length) * (sqrt(3) / 2)

In this case, the side length is equal to the side length of the equilateral triangle, which we calculated earlier as 8 cm * sqrt(3).

Substituting the value into the formula, we get:

Height = (8 cm * sqrt(3)) * (sqrt(3) / 2) = 12 cm

The area of an isosceles triangle can be calculated using the formula:

Area = (base length) * (height) / 2

In this case, the base length is equal to the side length of the equilateral triangle, which is 8 cm * sqrt(3).

Substituting the values into the formula, we get:

Area = (8 cm * sqrt(3)) * (12 cm) / 2 = 48 cm^2 * sqrt(3)

Since there are three identical triangular faces, the total lateral surface area is:

Total Lateral Surface Area = 3 * (48 cm^2 * sqrt(3)) = 144 cm^2 * sqrt(3)

Calculation of the Total Surface Area

To find the total surface area of the triangular pyramid, we need to sum up the base area and the lateral surface area.

Total Surface Area = Base Area + Lateral Surface Area

Substituting the values we calculated earlier, we get:

Total Surface Area = 16 cm^2 * sqrt(3) + 144 cm^2 * sqrt(3) = 160 cm^2 * sqrt(3)

Therefore, the total surface area of the triangular pyramid inscribed in the cone is 160 cm^2 * sqrt(3).

Conclusion

In conclusion, the surface area of the triangular pyramid inscribed in the cone with a height of 2 + √3 cm and a radius of the base of 4 cm is 160 cm^2 * sqrt(3).