Find the volume of a regular triangular pyramid with a base side of 6

Find the volume of a regular triangular pyramid with a base side of 6, the side edge of which is inclined to the base at an angle whose tangent is 1.5.

The volume of a regular triangular pyramid is equal to a third of the product of the base area by the height.

At the base of a regular triangular pyramid lies a regular triangle, all angles of which are 60 °. The area of ​​such a triangle is defined as half the product of the square of the side and the sine of the angle:

Sb = 0.5 * a ^ 2 * sin 60 ° = 0.5 * 6 ^ 2 * √3 / 2 = 9√3.

The projection of the side edge of the pyramid coincides with the radius of the circle described near the base-triangle, which can be found by the formula:

R = a / √3 = 6 / √3 = 2√3 – the projection of the lateral rib onto the base.

In a right-angled triangle, in which the lateral edge is the hypotenuse, the height and projection of the lateral edge are the legs, the height is the leg, opposite the angle between the lateral edge and the plane of the base, the tangent of which is 1.5. The ratio of the opposite leg to the adjacent leg is the tangent of the angle, which means:

h / R = tan α = 1.5;

h = 1.5 * R = 1.5 * 2√3 = 3√3 – the height of this pyramid.

V = Ssc * h / 3 = 9√3 * 3√3 / 3 = 27 – the required volume of the pyramid.