Question

Дана правильная четырехугольная пирамида ,высота равна 2 \sqrt{7},боковая грань 10,Найти площадь полной поверхности пирамиды

Answer 1

Пусть дана пирамида ABCDS, SO- высота пирамиды

AS=10

По теореме Пифагора

   (AO)^2=(AS)^2-(SO)^2=100-4/7=696/7

    AO=√(696/7)

    AC=2*AO=2√(696/7)

    Далее

    (AD)^2+(DC)^2=(AC)^2

    2*(AD)^2=2784/7 => (AD)^2=696/7

    AD=√(696/7)

Sосн=(√(696/7))^2=697/7

Пусть SK высота опущена с вершины S на AD, тогда AK=KD=(1/2)*(√696/7))=√174/7

   (SK)^2=(AS)^2-(AK)^2=100-174/7=693/7

    SK=√(693/7)

далее

    Sasd=(1/2)*KS*AD=(1/2)*√(693/7)*√(696/7)=√17226

 

S=Sосн +4Sasd=697/7+4*√17226

 

 

Answer 2

Calculation of the Surface Area of the Pyramid

To find the surface area of the given pyramid, we need to calculate the areas of all its faces and then sum them up.

Let's break down the calculation step by step:

1. Base Area: - Since the given pyramid is a regular quadrilateral pyramid, its base is a square. - The length of one side of the base can be calculated using the Pythagorean theorem, as the base is a right triangle with the hypotenuse being the side length of the base and the other two sides being half the length of the base and the height of the pyramid. - Let's denote the side length of the base as s. - Using the Pythagorean theorem, we have: - s^2 = (s/2)^2 + (2√7)^2 - Simplifying the equation, we get: - s^2 = s^2/4 + 28 - Multiplying both sides by 4, we get: - 4s^2 = s^2 + 112 - Rearranging the equation, we have: - 3s^2 = 112 - Solving for s, we find: - s = √(112/3) - Now that we have the side length of the base, we can calculate the area of the base by squaring the side length: base_area = s^2.

2. Lateral Faces: - The given pyramid has four lateral faces, each of which is an isosceles triangle. - The base of each lateral face is the side length of the base, s, and the height can be calculated using the Pythagorean theorem. - Let's denote the height of each lateral face as h. - Using the Pythagorean theorem, we have: - h^2 = (2√7)^2 - (s/2)^2 - Simplifying the equation, we get: - h^2 = 28 - s^2/4 - Substituting the value of s from the previous step, we can calculate h. - The area of each lateral face can be calculated as lateral_area = (1/2) * s * h. - Since there are four lateral faces, the total area of the lateral faces is 4 * lateral_area.

3. Total Surface Area: - The total surface area of the pyramid is the sum of the base area and the total area of the lateral faces. - Let's denote the total surface area as total_area. - We can calculate total_area as total_area = base_area + 4 * lateral_area.

Now, let's calculate the surface area of the given pyramid using the above steps and the given values:

1. Calculation of the base area: - s = √(112/3) - base_area = s^2

2. Calculation of the lateral faces: - h^2 = 28 - s^2/4 - lateral_area = (1/2) * s * h

3. Calculation of the total surface area: - total_area = base_area + 4 * lateral_area

Let's perform the calculations.