Question

В правильной четырехугольной пирамиде сторона основания равна 5 см, а высота 7 см. найти площадь поверхности пирамиды.

Answer 1

В правильной четырехугольной пирамиде сторона основания равна 5 см, а высота 7 см. найти площадь поверхности пирамиды.

Решение:

Поскольку четырехугольная пирамида правильная, то основанием пирамиды является квадрат. Тогда $\mathrm{OK=\dfrac{AD}{2}=\dfrac{5}{2}=2.5}$ см.

Из прямоугольного треугольника SOK вычислим апофему пирамиды SK:

$\mathrm{SK=\sqrt{OK^2+SO^2}=\sqrt{2.5^2+7^2}=\dfrac{\sqrt{221}}{2}}$ см

Площадь поверхности пирамиды: $\mathrm{S=S_{oc_H}+S_{bok}}$

Нужно найти площадь основания и площадь боковой поверхности пирамиды:

$\mathrm{S_{oc_H}=AD^2=5^2=25~_{CM^2}}$

$\mathrm{S_{bok}=\frac{1}{2}P_{oc_H}\cdot SK=\frac{1}{2}\cdot 4\cdot5\cdot\frac{\sqrt{221}}{2}=5\sqrt{221}} ~_{CM^2}}$

Окончательно получим: $\mathrm{S=(25+5\sqrt{221})}$ см²

Ответ: (25+5√221) см².

Answer 2

Calculation of the Surface Area of a Pyramid

To find the surface area of a pyramid, we need to know the length of the base and the height of the pyramid. In this case, the length of the base is given as 5 cm, and the height is given as 7 cm.

The formula for the surface area of a pyramid is:

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

Let's calculate the surface area step by step.

1. Calculate the base area: - The base of the pyramid is a square, so the base area is equal to the length of one side squared. - In this case, the length of the base is 5 cm, so the base area is 5 cm × 5 cm = 25 cm².

2. Calculate the perimeter of the base: - Since the base is a square, all sides are equal in length. - The perimeter of a square is equal to 4 times the length of one side. - In this case, the length of one side is 5 cm, so the perimeter of the base is 4 × 5 cm = 20 cm.

3. Calculate the slant height: - The slant height is the height of one of the triangular faces of the pyramid. - To find the slant height, we can use the Pythagorean theorem. - The slant height, the height of the pyramid, and half the length of the base form a right triangle. - Using the Pythagorean theorem, we can calculate the slant height as follows: - Slant Height = √(Height² + (1/2 × Base Length)²) - Slant Height = √(7 cm² + (1/2 × 5 cm)²) - Slant Height = √(49 cm² + 6.25 cm²) - Slant Height = √55.25 cm² - Slant Height ≈ 7.43 cm (rounded to two decimal places)

4. Calculate the surface area: - Using the formula mentioned earlier, we can now calculate the surface area of the pyramid. - Surface Area = Base Area + (1/2) × Perimeter of Base × Slant Height - Surface Area = 25 cm² + (1/2) × 20 cm × 7.43 cm - Surface Area = 25 cm² + 10 cm × 7.43 cm - Surface Area = 25 cm² + 74.3 cm² - Surface Area ≈ 99.3 cm² (rounded to one decimal place)

Therefore, the surface area of the given pyramid is approximately 99.3 cm².