# In a cylinder with a height of 10 cm, the axial section area is exactly 120 cm ^ 2, find: a) The radius of the base of the cylinder

**In a cylinder with a height of 10 cm, the axial section area is exactly 120 cm ^ 2, find: a) The radius of the base of the cylinder. b) the cross-sectional area of the parallel axis and spaced from it at a distance of 2 cm.**

Since the cylinder is formed by rotating a rectangle around its side, then its axial section is also a rectangle. For convenience, we will designate it as ABCD.

Since the area of a triangle is equal to the product of its length and width (S = AB ∙ BC), we can find the diameter of the base of this cylinder BC.

BC = S / AB;

BC = 120/10 = 12 cm.

Since the radius is half the diameter, then:

R = 12/2 = 6 cm.

To calculate the cross-sectional area of A1B1C1D1, it is necessary to multiply the height of A1B1 by the segment B1C1:

SА1В1С1Д1 = А1В1 ∙ В1С1.

In order to find the length of the segment B1C1, consider the base of the cylinder. Segments В1О and С1О are equal as radii. OH – perpendicular. Thus, we see that the triangles В1ОН and С1ОН are rectangular and equal to each other. Using the Pythagorean theorem, we find the length of the segment B1H.

B1O ^ 2 = OH ^ 2 + B1H ^ 2;

B1H2 = B1O ^ 2 – OH ^ 2;

B1H ^ 2 = 6 ^ 2 – 2 ^ 2 = 36 – 4 = 32;

В1Н = √32 ≈ 5.66 cm.

В1С1 = 5.66 ∙ 2 = 11, 32 cm.

SА1В1С1Д1 = 10 ∙ 11.32 = 113.2 cm2.

Answer: a) the radius of the base is 6 cm; b) the cross-sectional area is 113.2 cm2.