In a regular quadrangular pyramid, the height is 12 cm, and the height of the side edge is 15 cm. Find the volume of the pyramid.

univerkov | February 10, 2021 | education

Let’s connect the points O and Н.

The triangle MOН is rectangular, then, according to the Pythagorean theorem, OH ^ 2 = MH ^ 2 – MO ^ 2 = 225 – 144 = 81.

OH = 9 cm.

The side faces of the pyramid are equilateral triangles, then the height of MH is its median, and then DН = CH.

Point O is the middle of the AС diagonal, then OH is the middle line of the AСD triangle, and therefore AD = 2 * OH = 2 * 9 = 18 cm.

Determine the area of the base of the pyramid. Sbn = BP * СD = 18 * 18 = 324 cm2.

Then V = Sosn * MO / 3 = 324 * 12/3 = 1296 cm2.

Answer: The volume of the pyramid is 1296 cm2.

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