Question

Решить задачу основание стороны правильной треугольной призмы 6см а диагональ боковой грани 10см. Найти площадь боковой и полной поверхности призмы?

Answer 1

Рассмотрим боковую сторону: сторона а=6 (сторона треугольника), диагональ с=10, второая сторона б находится то теореме Пифагора: б=√(с²-а²)=√(100-36)=√64=8 (см)
Площадь боковой поверхности равна площади боковой стороны, умноженной на 3:
Sбок=а*б*3=6*8*3=144 (см²)
Sполн.=Sбок+2*Socн.
S оснований находится по формуле площадт равностороннего треугольника (призма - правильная): Sосн=а²√3/4=36*√3/4=9√3 (см²)
Отсюда:
Sполн=144+18√3=18(8+√3) (cм²)
Ответ: Sбок=144 см²; Sполн=18(8-√3) см²

Answer 2

Problem Analysis

We are given the base side length of a right triangular prism as 6 cm and the diagonal of one of the lateral faces as 10 cm. We need to find the lateral surface area and the total surface area of the prism.

Solution

To find the lateral surface area of the prism, we need to calculate the area of the three rectangular lateral faces. The lateral surface area can be calculated using the formula:

Lateral Surface Area = Perimeter of Base x Height

The base of the prism is a right triangle with side lengths 6 cm, 8 cm, and 10 cm (using the Pythagorean theorem). The perimeter of the base can be calculated as the sum of the three sides:

Perimeter of Base = 6 cm + 8 cm + 10 cm

The height of the prism can be calculated as the length of the altitude from one of the vertices of the base triangle to the opposite face. Since the diagonal of the lateral face is given as 10 cm, we can use the Pythagorean theorem to find the height:

Height = √(Diagonal^2 - Base Side^2) = √(10 cm^2 - 6 cm^2)

Once we have the perimeter of the base and the height, we can calculate the lateral surface area.

To find the total surface area of the prism, we need to add the areas of the two triangular bases to the lateral surface area. The area of each triangular base can be calculated as half the product of the base and the corresponding altitude. Since the base is a right triangle, the altitude can be calculated as the length of the perpendicular from the right angle vertex to the hypotenuse. Using the base side length of 6 cm, we can calculate the altitude as:

Altitude = Base Side / 2 = 6 cm / 2

Once we have the lateral surface area and the areas of the two triangular bases, we can calculate the total surface area.

Let's calculate the lateral surface area and the total surface area of the prism.

Calculation

Given: Base side length = 6 cm Diagonal of lateral face = 10 cm

Using the Pythagorean theorem, we can find the other two sides of the base triangle: Side 1 = 6 cm Side 2 = 8 cm Side 3 (hypotenuse) = 10 cm

Perimeter of Base = Side 1 + Side 2 + Side 3 = 6 cm + 8 cm + 10 cm = 24 cm

Height = √(Diagonal^2 - Base Side^2) = √(10 cm^2 - 6 cm^2) = √(100 cm^2 - 36 cm^2) = √(64 cm^2) = 8 cm

Lateral Surface Area = Perimeter of Base x Height = 24 cm x 8 cm = 192 cm^2

Altitude of Base = Base Side / 2 = 6 cm / 2 = 3 cm

Area of each Triangular Base = (Base Side x Altitude) / 2 = (6 cm x 3 cm) / 2 = 9 cm^2

Total Surface Area = Lateral Surface Area + 2 x Area of Triangular Base = 192 cm^2 + 2 x 9 cm^2 = 192 cm^2 + 18 cm^2 = 210 cm^2

Answer

The lateral surface area of the prism is 192 cm^2 and the total surface area of the prism is 210 cm^2.