Question

Диагональ боковой грани правильной четырехугольной призмы равна 15см, а сторона основания равна 9 см. Найти полную поверхность призмы.

Answer 1

Пусть боковая грань   x  тогда 

9^2+x^2=15^2

x^2=15^2-9^2

x=12

 

Значит площади основания равны 2S=2*9^2 =  162        

 и боковые 4S =4*12*9=432

S полная   = 432+162   =594  см кв 

 

 

 

Answer 2

Призма АВСА1В1С1 (АВС у меня верхняя плоскость)

рассм тр АС1С:

АС1=15(по ус)

СС1=9(по ус)

по теореме Пифагора находим АС

АС=Корень(15^2-9^2)=12

Sбок=12*3*9=324 см^2

рассм тр АВС

он равносторонний,

стороны=12

чтобы найти его площадь проведем высоту ВН

рассм тр ВНС

он прямоугольный,

ВС=12

НС=12/2=6(в правильном треугольнике высота является медианой, значит АН=НС)

по теореме Пифагора найдём ВН

ВН=корень(ВС-НС)=корень(144-36)=корень(108)=6*корень(3)

SАВС=1/2АС*ВН=36*корень(3)

S пол=2*36*корень(3)+324=72*корень(3)+324

Answer 3

Calculation of the Surface Area of a Prism

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

Given information: - Diagonal of the lateral face of the prism = 15 cm - Side length of the base = 9 cm

To calculate the surface area, we need to find the areas of the lateral faces and the base.

Finding the Area of the Lateral Face

The lateral face of a prism is a rectangle. To find its area, we need to know the length and width of the rectangle.

The length of the lateral face can be found using the diagonal and the side length of the base. The diagonal of a rectangle divides it into two congruent right triangles. The side length of the base is one of the legs of these triangles.

Using the Pythagorean theorem, we can find the length of the lateral face:

Length of the lateral face = √(diagonal^2 - base side^2)

Substituting the given values: Length of the lateral face = √(15^2 - 9^2)

Now, let's calculate the length of the lateral face.

Calculation of the Length of the Lateral Face

Length of the lateral face = √(15^2 - 9^2) Length of the lateral face = √(225 - 81) Length of the lateral face = √144 Length of the lateral face = 12 cm

Calculation of the Area of the Lateral Face

The area of a rectangle can be calculated by multiplying its length and width. In this case, the width of the lateral face is the same as the side length of the base.

Area of the lateral face = length of the lateral face * side length of the base

Substituting the given values: Area of the lateral face = 12 cm * 9 cm

Now, let's calculate the area of the lateral face.

Calculation of the Area of the Lateral Face

Area of the lateral face = 12 cm * 9 cm Area of the lateral face = 108 cm^2

Calculation of the Area of the Base

The base of the prism is a square, so its area can be calculated by squaring the side length.

Area of the base = side length of the base^2

Substituting the given value: Area of the base = 9 cm^2

Now, let's calculate the area of the base.

Calculation of the Area of the Base

Area of the base = 9 cm^2

Calculation of the Total Surface Area

To find the total surface area, we need to sum the areas of the lateral faces and the base.

Total Surface Area = 2 * Area of the lateral face + Area of the base

Substituting the calculated values: Total Surface Area = 2 * 108 cm^2 + 9 cm^2

Now, let's calculate the total surface area.

Calculation of the Total Surface Area

Total Surface Area = 2 * 108 cm^2 + 9 cm^2 Total Surface Area = 216 cm^2 + 9 cm^2 Total Surface Area = 225 cm^2

Answer

The total surface area of the given prism is 225 cm^2.

Please let me know if there is anything else I can help you with.