In the quadrangular pyramid SABCD, the length of each edge is 18 cm. The section plane passes through

| June 18, 2021 | education

In the quadrangular pyramid SABCD, the length of each edge is 18 cm. The section plane passes through the diagonal BD and is perpendicular to the edge SC. Calculate the volume of the pyramid, the top of which is point S, and the base is the section of the given pyramid.

Since the section is perpendicular to the edge SC, the height of the resulting SDBH pyramid will be equal to SH.

The volume of the pyramid is equal to one third of the product of the area of ​​the base and the height.

V = 1/3 * S main * h.

Since the pyramid is correct, all the edges are 18 cm, then the heights BH, OH and DH will also be the medians, that is, the height of the pyramid SH = 1/2 * SC = 9 cm.

The base area of ​​the SDBH pyramid is the cross-sectional area.

Sbn = 1/2 * OH * BD.

Triangle ABD is rectangular (since ABCD is a square), according to the Pythagorean theorem:

BD² = AB² + BD² = 18² + 18² = 2 * 18².

BD = √ (2 * 18²) = 18√2 (cm).

Triangle OСН – rectangular (section perpendicular to SC), OC = 1/2 * AC = 1/2 * BD = 8√2 cm, СН = 9 cm, according to the Pythagorean theorem:

OH² = OC² – CH² = (9√2) ² – 9² = 81 * 2 – 81 = 81.

OH = √81 = 9 (cm).

Hence, Ssc = 1/2 * 9 * 18√2 = 81√2 (cm²).

Therefore, the volume of the pyramid is:

V = 1/3 * 9 * 81√2 = 243√2 (cm3).



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