The height of a regular quadrangular pyramid is 6 cm. It forms an angle of 30 degrees with

| August 17, 2021 | education

The height of a regular quadrangular pyramid is 6 cm. It forms an angle of 30 degrees with the side face. find the volume of the pyramid.

At the base of a regular quadrangular pyramid lies a square, the height of the pyramid from its top drops to the center of the base, which is the point of intersection of the diagonals. The height is perpendicular to the base, we have a right-angled triangle formed by the height of the pyramid, the height of the side edge and the projection of the height of the side edge onto the base, equal to half of the side of the base.
The ratio of the opposite leg to the adjacent leg is the tangent of the angle, therefore, half of the side of the base of the pyramid can be found as the product of the height and the tangent of the angle formed by the height and the side face:
a / 2 = h * tg30;
a = 2 * h * tg30 = 2 * 6 * √3 / 3 = 4√3 cm.
The area of ​​the base of the pyramid is equal to the square of the side of the base: Sbase = a ^ 2 = 16 * 3 = 48 cm2.
The volume of the pyramid is equal to a third of the product of its height by the area of ​​the base: V = 1/3 * Sbn * h = 48 * 6/3 = 96 cm3.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.