Question

В правильной треугольной пирамиде боковое ребро равно 6 и составляет с плоскостью основания угол 60,найдите площадь поверхности пирамиды

Answer 1

Мне на ум приходит только один способ, самый элементарный и бездарный... Плоскость каждой грани получается равносторонним треугольником. Надо взять одну грань и рассмотреть как треугольник...

Построим в равностороннем треугольнике ABC высоту. Теперь через пифагора:
36 = 9 + $x^2$
$X^2$ = 25
X = 5

 Высота = 5. Площадь тр-ка ABC - полупроизведение основания на высоту, или:
$S _{ABC}$ = 5 * 6/2 = 5*3 = 15 cм^2

У треугольной пирамиды в общей сложности четыре грани. Значит, её грани - четыре одинаковых треугольника, и площадь поверхности такой пирамиды будет равна учетверённой площади одной грани, или:
$S_{pyramid}$ = 4 * $S_{triangle}$ = 4*15 = 60


Вроде так)

Answer 2

Finding the Surface Area of a Right Triangular Pyramid

To find the surface area of a right triangular pyramid, we can use the formula:

Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height

Where: - Base Area is the area of the base triangle. - Perimeter of Base is the sum of the lengths of the base triangle's sides. - Slant Height is the height of one of the triangular faces from the apex to the base.

Given Information

In this case, we are given: - The length of the side of the base triangle, which is 6. - The angle between the side of the pyramid and the base plane, which is 60 degrees.

Calculating the Surface Area

First, let's calculate the slant height of the pyramid using the given information.

The slant height (l) can be calculated using the formula: l = a / sin(θ)

Where: - a is the length of the side of the base triangle. - θ is the angle between the side of the pyramid and the base plane.

Using the given values: - a = 6 - θ = 60 degrees

We can calculate the slant height: l = 6 / sin(60°)

Using the trigonometric identity sin(60°) = √3 / 2: l = 6 / (√3 / 2) l = 12 / √3 l = 4√3

Now, we have the slant height (l) as 4√3.

Next, we need to calculate the base area and the perimeter of the base triangle to find the surface area.

Base Area and Perimeter of Base

The base triangle is a right-angled triangle with sides of length 6, 3, and 3 (using the Pythagorean theorem).

The base area (A) can be calculated using the formula: A = (1/2) * base * height

Where: - base = 6 - height = 3 (half of the base)

A = (1/2) * 6 * 3 A = 9

The perimeter of the base triangle (P) is the sum of the lengths of its sides: P = 6 + 3 + 3 P = 12

Calculating the Surface Area

Now, we can use the formula for the surface area of the pyramid: Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height

Substituting the calculated values: Surface Area = 9 + (1/2) * 12 * 4√3 Surface Area = 9 + 6 * 4√3 Surface Area = 9 + 24√3

So, the surface area of the given right triangular pyramid is 9 + 24√3.