Question

В правильной треугольной пирамиде сторона основания равна 4 корней из 3, боковое ребро 5 см. Найдите объем и площадь боковой поверхности пирамиды.

Answer 1

Vпир=(1/3)*Sосн*h
Sбок=(1/2)*Pосн*l, где Sосн - площадь основания (правильный треугольник), h - высота пирамиды, Pосн - периметр основания, l - апофема=боковое ребро пирамиды.

Sосн=$\frac{1}{2}*4 \sqrt{3}* 4\sqrt{3}*sin60=24* \frac{ \sqrt{3} }{2}=12 \sqrt{3}$

Sбок=$\frac{1}{2}*(4 \sqrt{3}*3)*5=30 \sqrt{3}$

Найдём объём пирамиды. Пусть SABC - пирамида, SO=h - её высота. Проведём СМ - высоту в равностороннем треугольнике основания (она также будет являться медианой) и медиану BL. Тогда точка O окажется в точке пересечения медиан. Медианы точкой пересечения делятся в отношении 2:1, считая от вершины, то есть CO=(2/3)CM. Из прямоугольного треугольника CMB найдём $CM=BC*sin60=4 \sqrt{3}* \frac{ \sqrt{3}}{2}=6$. Тогда $OC= \frac{2}{3}*6=4$. По теореме Пифагора в ΔSOC: $h=SO= \sqrt{25-16}=3$. $V= \frac{1}{3}*12 \sqrt{3}*3=12 \sqrt{3}.$

Answer 2

Calculation of Volume and Surface Area of a Right Triangular Pyramid

To find the volume and surface area of a right triangular pyramid, we need to know the length of the base and the length of one of the lateral edges.

Given information: - Length of the base of the pyramid = 4√3 cm - Length of one of the lateral edges = 5 cm

Volume of the Pyramid

The volume of a pyramid can be calculated using the formula:

Volume = (1/3) * Base Area * Height

In this case, the base of the pyramid is a right triangle, so the base area can be calculated using the formula:

Base Area = (1/2) * Base Length * Height

Since the base is a right triangle, we can use the Pythagorean theorem to find the height of the triangle.

Let's calculate the height of the triangle first:

Height = √(Lateral Edge^2 - (Base Length/2)^2)

Substituting the given values:

Height = √(5^2 - (4√3/2)^2)

Simplifying the equation:

Height = √(25 - (16*3/4)) = √(25 - 12) = √13

Now, we can calculate the base area:

Base Area = (1/2) * Base Length * Height = (1/2) * 4√3 * √13 = 2√3 * √13 = 2√39

Finally, we can calculate the volume of the pyramid:

Volume = (1/3) * Base Area * Height = (1/3) * 2√39 * √13 = (2/3) * √507

Therefore, the volume of the pyramid is (2/3) * √507 cubic centimeters.

Surface Area of the Pyramid

The surface area of a pyramid can be calculated by adding the areas of the base and the lateral faces.

The base area has already been calculated as 2√39.

To calculate the area of each lateral face, we need to find the slant height of the pyramid. The slant height can be calculated using the Pythagorean theorem:

Slant Height = √(Height^2 + (Base Length/2)^2)

Substituting the given values:

Slant Height = √(√13^2 + (4√3/2)^2) = √(13 + 12) = √25 = 5

Since the base is a right triangle, the lateral faces are also right triangles. Therefore, the area of each lateral face can be calculated using the formula:

Lateral Face Area = (1/2) * Base Length * Slant Height

Substituting the given values:

Lateral Face Area = (1/2) * 4√3 * 5 = 10√3

Now, we can calculate the surface area of the pyramid:

Surface Area = Base Area + 4 * Lateral Face Area = 2√39 + 4 * 10√3 = 2√39 + 40√3

Therefore, the surface area of the pyramid is 2√39 + 40√3 square centimeters.

Please note that the calculations provided above are based on the given information and assumptions.