Question

Дана правильная треугольная пирамида PABC, ребро основания которой 8 см, апофема пирамиды - 3 см. Найдите площадь боковой поверхности пирамиды. ​

Answer 1

Ответ: площадь боковой поверхности правильной треугольной пирамиды равна 36 см².

Пошаговое объяснение:

Площадь боковой поверхности правильной пирамиды равна половине произведения периметра основания и апофемы: Sбок = 1/2 · Росн · h, где Sбок - площадь боковой поверхности, Росн - периметр основания, h - апофема (высота боковой грани).

По условию: сторона основания а = 8 см, h = 3 см.

Т.к. пирамида правильная треугольная, то Росн = 3а = 3 · 8 = 24 (см).

Sбок = 1/2 · 24 · 3 = 36 (см²).

Answer 2

Calculation of the lateral surface area of a pyramid

To find the lateral surface area of a pyramid, we need to know the length of the base edge and the apothem of the pyramid. In this case, we are given that the base edge of the pyramid is 8 cm and the apothem is 3 cm.

The formula to calculate the lateral surface area of a pyramid is:

Lateral Surface Area = (Perimeter of the base) x (Apothem) / 2

To find the perimeter of the base, we need to know the length of one side of the base. Since the given pyramid is a right triangular pyramid, we can use the Pythagorean theorem to find the length of one side of the base.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the hypotenuse is the base edge of the pyramid, which is 8 cm, and the apothem is the height of the right triangle, which is 3 cm. Let's calculate the length of one side of the base using the Pythagorean theorem:

Base edge = 8 cm

Apothem = 3 cm

Side of the base = √(Base edge^2 - Apothem^2)

Side of the base = √(8^2 - 3^2) = √(64 - 9) = √55 ≈ 7.42 cm

Now that we know the length of one side of the base, we can calculate the perimeter of the base:

Perimeter of the base = 3 x (Side of the base) = 3 x 7.42 cm ≈ 22.26 cm

Finally, we can calculate the lateral surface area of the pyramid using the formula mentioned earlier:

Lateral Surface Area = (Perimeter of the base) x (Apothem) / 2

Lateral Surface Area = 22.26 cm x 3 cm / 2 ≈ 33.39 cm²

Therefore, the lateral surface area of the given pyramid is approximately 33.39 cm².