Question

В правильной треугольной пирамиде стороны основания = a, а двугранный угол при ребре основания альфа. найти площадь поверхности шара, вписанного в пирамиду

Answer 1

АВС -осноание пирамиды Д -вершина ДО высота пирамиды АР- высота основания на ВС РО=2а АР=3РО=6а R=2РО=4а -радиус описанной окр. вокруг основания АВ=√3R=4√3a h=РОtgα=2atgα V=Sh/3 S=0.5BC*AP=0.5(4√3a)(6а)=12√3a² V=(12√3a²)(2atgα)/3=8√3a³tgα

Answer 2

Finding the Surface Area of a Sphere Inscribed in a Right Triangular Pyramid

In a right triangular pyramid with base sides of length a and a dihedral angle of α at the base edge, we can calculate the surface area of the sphere inscribed in the pyramid.

To find the surface area of the inscribed sphere, we can use the formula:

S = 2πr^2 + πdL

Where: - S is the surface area of the sphere - r is the radius of the inscribed sphere - d is the slant height of the pyramid - L is the length of the base edge of the pyramid

Calculating the Radius of the Sphere

The radius of the inscribed sphere can be calculated using the formula:

r = (V / (3√3))^(1/3)

Where: - V is the volume of the pyramid

Calculating the Volume of the Pyramid

The volume of the right triangular pyramid can be calculated using the formula:

V = (1/3) * A * h

Where: - A is the area of the base triangle - h is the height of the pyramid

Calculating the Area of the Base Triangle

The area of the base triangle can be calculated using the formula:

A = (1/2) * a^2 * sin(α)

Where: - a is the length of the base side - α is the dihedral angle at the base edge

Calculating the Height of the Pyramid

The height of the pyramid can be calculated using the formula:

h = (a/2) * cot(α/2)

Where: - a is the length of the base side - α is the dihedral angle at the base edge

Calculating the Slant Height of the Pyramid

The slant height of the pyramid can be calculated using the formula:

d = √(h^2 + (a/2)^2)

Where: - h is the height of the pyramid - a is the length of the base side

By using these formulas, you can calculate the surface area of the sphere inscribed in the right triangular pyramid.