The radius of the base of the cylinder is 13 cm, its height is 20 cm. Find the cross-sectional area of the

The radius of the base of the cylinder is 13 cm, its height is 20 cm. Find the cross-sectional area of the cylinder drawn parallel to the axis at a distance of 5 cm from it.

As you can see in the figure, the section parallel to the cylinder axis has the shape of a rectangle. Therefore, to calculate its area, it is necessary to multiply its length AB, which is equal to the height of the cylinder, by the width, which is the chord of the base of the BC.

The BO and OC segments are the radii of the base, therefore, they have the same length of 13 cm. Thus, we see that the triangle ΔBOC is isosceles, and the distance from the section to the axis is its height OH.

The BC chord is equal to the sum of the BH and HC segments;

ВС = ВН + НС.

Using the Pythagorean theorem, we find the BH segment:

BO ^ 2 = BH ^ 2 + OH ^ 2;

BH ^ 2 = BO ^ 2 – OH ^ 2;

BH ^ 2 = 13 ^ 2 – 5 ^ 2 = 169 – 25 = 144;

BH = √144 = 12 cm.

BC = 12 2 = 24 cm.

Now let’s find the cross-sectional area:

S = AB · BC;

S = 20 24 = 480 cm2.

Answer: the cross-sectional area of ​​the AVSD is 480 cm2.