The chord, which lies at the base of the cylinder, is 3√3 cm and contracts the arc of 120 degrees.

univerkov | August 15, 2021 | education

The chord, which lies at the base of the cylinder, is 3√3 cm and contracts the arc of 120 degrees. The line that connects one end of the chord to the center of the other base creates an angle of 45 degrees to the base plane. Find the total surface area of the cylinder.

Let us draw segments OS and OD to the edges of the chord CD. Since ОD = OC = R, the triangle COD is isosceles.

From point O let us lower the height OH to the chord CD.

Since the triangle is isosceles, the height OH divides the chord in half and the angle COD also divides it in half.

CH = DH = CD / 2 = 3 * √3 / 2 cm.

The chord CD contracts the arc 120, then the central angle СОD = 120 then the angle СОН = 120/2 = 600.

In the right-angled triangle SON, we determine the length of the hypotenuse CO.

CO = DO CH / Sin60 = (3 * √3 / 2) / √3 / 2 = 3 cm.

Since in triangle OO1D the angle ODO1 = 45, then triangle OO1D is rectangular and isosceles. OO1 = DO = 3 cm.

Determine the area of the base.

Sop = n * R2 = n * 32 = n * 9 m2.

Let us determine the area of the lateral surface.

Side = 2 * n * R * OO1 = 2 * n * 3 * 3 = n * 18 cm2.

Let us determine the total surface area of the cylinder.

S floor = 2 * S main + S side = 2 * n * 9 + 2 * 18 = n * 36 cm2.

Answer: The total surface area of the cylinder is n * 36 cm2.

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