In a regular triangular pyramid, the lateral rib is inclined to the base plane at an angle of 60 °.

In a regular triangular pyramid, the lateral rib is inclined to the base plane at an angle of 60 °. A plane is drawn through the side of the base at an angle of 30 ° to the plane of the base. Find the area of the pyramid with this plane if the side of the base is 12.

First, let’s draw a regular triangular SABC pyramid.

Let’s draw a section of the ВСК through the side of the BC at an angle of 30 ° to the base.

Then the side KВ = KС, then we draw the height KM, then the angle KMA = 30.

The angle SMA = 60, then the angle MKA = 90o, therefore KM = AM * cos30 °.

MA = AC * sin60о = 6√3, then MK = 9 cm.

Therefore, the area is ½ * 9 * 12 = 54 cm2.