Question

В правильной треугольной призме через сторону основания нижнего основания и противолежащую ей вершину верхнего основания проведено сечение, составляющее угол в 60 градусов с плоскостью основания. Найдите объем призмы , если сторона основания равна а. с рисунком

Answer 1

BH_|_AC,<B1HB=60,<A=<B=<C=60
BH=BC*sin<C
BH=a√3/2
B1H=BH/cos<B1HB
B1H=a√3/2:1/2=a√3
Sсеч=1/2B1H*AC
S=1/2*a√3*a=a²√3/2

Answer 2

Finding the Volume of a Right Triangular Prism

To find the volume of a right triangular prism, we need to know the length of the base side and the height of the prism. In this case, the base side is given as "a."

The volume of a prism is calculated by multiplying the area of the base by the height. Since the base of the prism is a right triangle, we can use the formula for the area of a right triangle to find the base area.

The formula for the area of a right triangle is:

Area = (1/2) * base * height

In this case, the base of the right triangle is "a" and the height is the distance between the base and the opposite vertex of the upper base.

To find the height of the prism, we can use trigonometry. Since the angle between the base and the cutting plane is given as 60 degrees, we can use the sine function to find the height.

The formula for the height of the prism is:

Height = sin(60) * a

Now that we have the base area and the height, we can calculate the volume of the prism using the formula:

Volume = base area * height

Let's substitute the values into the formulas and calculate the volume.

Step 1: Calculate the base area: The base area of the right triangle is given by the formula (1/2) * base * height.

Base Area = (1/2) * a * a * sin(60) Step 2: Calculate the height: The height of the prism is given by the formula sin(60) * a.

Height = sin(60) * a Step 3: Calculate the volume: The volume of the prism is given by the formula base area * height.

Volume = (1/2) * a * a * sin(60) * sin(60) * a

Simplifying the expression:

Volume = (1/2) * a^2 * sin^2(60) * a

Since sin^2(60) is equal to 3/4, we can substitute this value:

Volume = (1/2) * a^2 * (3/4) * a

Simplifying further:

Volume = (3/8) * a^3

Therefore, the volume of the right triangular prism is (3/8) * a^3.

I hope this helps! Let me know if you have any further questions.