Question

Площадь боковой поверхности правильной треугольной пирамиды 63 см^2, сторона основания-2 см.Найдите объм пирамиды.

Answer 1

Проекция апофемы SД на основание равна (1/3) высоты h треугольника в основании пирамиды - это отрезок ОД.
h = a*cos30° = 2*√3/2 = √3.
ОД = (1/3)*√3 = 1/√3.
Длину апофемы находим из выражения Sбок = (1/2)*А*Р.
Периметр Р = 3*2 = 6 см.
А = 2S/P = 2*63/6 = 21 см.
Высота пирамиды Н = √(А²-ОД²) = √(441-(1/3)) = 
= √(1322/3) = √440,6667 = 20,99206.
Площадь основания равна:
So = a²√3/4 = 4*√3/4 = √3 см².
Объём пирамиды равен:
V = (1/3)So*H = (1/3)*√3*20,99206 =  12,11977 см³.

Answer 2

Calculation of the Volume of a Triangular Pyramid

To find the volume of a triangular pyramid, we need to know the area of the base and the height of the pyramid. In this case, we are given the area of the lateral surface of the pyramid, which is 63 cm^2, and the side length of the base, which is 2 cm.

To calculate the volume, we can use the formula:

Volume = (1/3) * Base Area * Height

Since the base of the pyramid is a triangle, we can find the area of the base using the formula for the area of a triangle:

Base Area = (1/2) * Base Length * Height of the Base

In this case, the base length is 2 cm, and we need to find the height of the base.

To find the height of the base, we can use the formula for the area of a triangle:

Base Area = (1/2) * Base Length * Height of the Base

Given that the area of the lateral surface of the pyramid is 63 cm^2, we can find the height of the base using the formula:

Height of the Base = (2 * Area of the Lateral Surface) / Base Length

Let's calculate the height of the base first:

Height of the Base = (2 * 63 cm^2) / 2 cm = 63 cm

Now that we have the height of the base, we can calculate the base area:

Base Area = (1/2) * Base Length * Height of the Base = (1/2) * 2 cm * 63 cm = 63 cm^2

Finally, we can calculate the volume of the pyramid:

Volume = (1/3) * Base Area * Height = (1/3) * 63 cm^2 * 63 cm = 1323 cm^3

Therefore, the volume of the given triangular pyramid is 1323 cm^3.

Please note that the calculations above are based on the given information and assumptions about the shape of the pyramid. If there are any additional details or constraints, please let me know, and I'll be happy to assist you further.