In the cylinder, the perimeter of the axial section is 40 cm, the diagonal of this section forms an angle of 45 degrees with the base plane. Find the volume of the cylinder.
Since the cylinder is formed as a result of the rotation of a rectangle around its side, its axial section also has the shape of a rectangle. For convenience, we will designate it as ABCD.
The AC diagonal of this section divides it into two equal right-angled triangles.
Consider a triangle ΔАСD. Since the angle ∠АDС is a straight line, the angle ∠САD is 45º, and the sum of all angles of the triangle is 180º, then:
∠АСD = 180º – ∠ADС – ∠САD;
∠АСD = 180º – 90º – 45º = 45º.
From this we see that this triangle is isosceles, in which AD = CD.
AD = CD = AB = BC.
Since the perimeter of the axial section is 40, and all its four sides are equal, then:
AD = CD = AB = BC = 40/4 = 10 cm.
Thus, the height of the cylinder is equal to the sides AB, CD and is 10 cm, and the diameter is equal to the sides of BC, AD and is also 10 cm.
The base radius is equal to half of its diameter:
r = d / 2;
r = 10/2 = 5 cm.
The volume of a cylinder is equal to the product of its base area by its height:
V = Sb. H = πr ^ 2h;
V = 3.14 * 5 ^ 2 * 10 = 3.14 * 25 * 10 = 785 cm3.
Answer: the volume of the cylinder is 785 cm3.