Pyramid and Cone

1. Characterize the calculation of volume and surface area for:

Solution:

2. A regular square pyramid is given (the base is a square with side a).

Fill in the missing values of the table.

Solution:

3. Above each face of a cube with edge a = 30 cm, a regular square pyramid with height 15 cm is constructed.

Solution:



The volume of the solid in the first case is V = 54 dm³, in the second case it is zero.

4. Calculate the volume of a pyramid whose lateral edge of length 5 cm forms an angle α = 60° with the square base. (Angle α is the angle between the edge and the diagonal of the base.)

Solution:


The volume of the pyramid is V = 18.04 cm³.

5. Determine the mass of a concrete pillar (ρ = 2.2 g.cm^-3) in the shape of a regular square frustum of a pyramid, if its square bases have sides a = 45 cm, b = 25 cm, and the height of the pillar is v = 33 cm.

Solution:



The mass of the concrete pillar is m = 91.355 kg.

6. A cone with dimensions given in the table is given.

Fill in the table.

Solution:

7. A right triangle with legs a = 3 cm, b = 4 cm rotates around the longer leg.

Solution:


The cone has volume V = 37.68 cm³ and surface area S = 75.36 cm².

8. The surface area of a cone is S = 235.5 cm². The axial section of the cone is an equilateral triangle.

Solution:



The volume of the cone is 226.6 cm³.

9. The lateral surface of a cone, developed into the plane, has the shape of a circular sector with central angle α = 150° and area S = 523.4 cm².

Solution:

S = 523.4 cm² – area of the circular sector with radius R – lateral surface area with radius r
o = R · arcα – circular arc corresponding to the sector
O = 2πr – circumference of the base circle of the cone



The dimensions of the cone are r = 8.33 cm, s = 20 cm, v = 18.18 cm and volume V = 1320.4 cm³.

10. The surface area of a frustum of a cone is 7693 cm², the radii of the bases are 28 cm and 21 cm.

Solution:

S = 7693 cm²
R = 28 cm
r = 21 cm



The height of the cone is v = 24 cm, its volume V = 45.5 dm³.

11. The volume of a frustum of a cone is V = 38 000π cm³. The radius of the lower base is 10 cm larger than the radius of the upper base.

Solution:



The radii of the bases of the frustum of the cone are R = 30 cm and r = 20 cm.

12. A frustum of a cone with radii x = 15 cm, y = 13 cm and height v = 9 cm was rolled out into a cylinder with radius r = 7.67 cm.

Solution:



The length / height of the cylinder is 30 cm.