The diagonal of the axial section of the cylinder is 82 dm and makes an angle of 45 degrees with the plane

The diagonal of the axial section of the cylinder is 82 dm and makes an angle of 45 degrees with the plane of the base of the cylinder. Find the total surface area of the cylinder.

The diagonal section of the cylinder is the rectangle ABCD, then the triangle ACD is rectangular, in which one of the acute angles is 45, then this triangle is isosceles, AD = CD.

Then AD = CD = AC * Sin45 = 82 * √2 / 2 = 41 * √2 dm.

Determine the area of the base of the cylinder.

Sb = π * АD ^ 2/4 = π * (41 * √2) ^ 2/4 = π * 840.5 dm2.

Let us determine the area of the lateral surface.

Side = π * AD * AB = π * 41 * √2 * 41 * √2 = π * 3362 dm2.

Then Sпов = 2 * Sсн + Sbok = 2 * π * 840.5 + π * 3362 = π * 5043 dm2.

Answer: The total surface area is π * 5043 dm2.