Question

Сторона основания правильной треугольной пирамиды 6 см, а ее боковые ребра равны 8 см. Найти площадь полной поверхности и объем пирамиды.

Answer 1

Sполн = Sбок + Sосн
Sбок = 1/2Р * L, где L - апофема
По теореме Пифагора L = корень из (64 - 9) = корень из 55
Sбок = 1/2 * 18 * V55 = 9V55
Sосн = 1/2 6*6 sin 60 = 9V3
Sполн = Sбок + Sосн = 9V55 + 9V3 = 9(V55 + V3)
V = 1/3Sосн * h
По теореме Пифагора h = корень из(64 - 12) = корень из 52 = 2V13
V = 1/3 * 9V3 * 2V13 = 6V39

Answer 2

Given Information

We are given that the base of a right triangular pyramid has a side length of 6 cm, and the lateral edges are equal to 8 cm.

Finding the Surface Area

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

# Area of the Base

The base of the pyramid is a right triangle with side length 6 cm. The formula to find the area of a right triangle is (base * height) / 2. In this case, the base and height are both 6 cm, so the area of the base is:

Area of the base = (6 cm * 6 cm) / 2 = 18 cm²

# Area of the Lateral Faces

The lateral faces of the pyramid are triangles. To find the area of each lateral face, we need to calculate the height of the triangle.

Let's denote the height of the triangle as 'h'. We can use the Pythagorean theorem to find the height. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the hypotenuse is the lateral edge, which is 8 cm, and one of the other sides is the height 'h'. The remaining side is the base of the triangle, which is 6 cm. Applying the Pythagorean theorem, we have:

8² = 6² + h² 64 = 36 + h² h² = 64 - 36 h² = 28 h = √28 ≈ 5.29 cm

Now that we have the height of the triangle, we can calculate the area of each lateral face using the formula (base * height) / 2. The base of each lateral face is the side length of the base, which is 6 cm. Therefore, the area of each lateral face is:

Area of each lateral face = (6 cm * 5.29 cm) / 2 ≈ 15.87 cm²

Since there are four lateral faces, the total area of the lateral faces is:

Total area of the lateral faces = 4 * 15.87 cm² = 63.48 cm²

# Total Surface Area

The total surface area of the pyramid is the sum of the area of the base and the area of the lateral faces. Therefore, the total surface area is:

Total surface area = Area of the base + Total area of the lateral faces = 18 cm² + 63.48 cm² = 81.48 cm²

Finding the Volume

To find the volume of the pyramid, we need to calculate the volume of the base and multiply it by the height of the pyramid.

# Volume of the Base

The base of the pyramid is a right triangle with side length 6 cm. The formula to find the area of a right triangle is (base * height) / 2. In this case, the base and height are both 6 cm, so the area of the base is:

Area of the base = (6 cm * 6 cm) / 2 = 18 cm²

# Volume of the Pyramid

To find the volume of the pyramid, we multiply the area of the base by the height of the pyramid. The height of the pyramid is the height of the lateral face, which we calculated earlier as approximately 5.29 cm.

Volume of the pyramid = Area of the base * Height = 18 cm² * 5.29 cm ≈ 95.22 cm³

Answer

Therefore, the surface area of the pyramid is approximately 81.48 cm² and the volume of the pyramid is approximately 95.22 cm³.