In a regular truncated quadrangular pyramid, the diagonal is 18, the side length of the lower base is 14

| September 22, 2021 | education

In a regular truncated quadrangular pyramid, the diagonal is 18, the side length of the lower base is 14, and the side length of the upper base is 10. Find the truncated volume. pyramids.

Let’s define the diagonals of the bases of the truncated pyramid.

AC ^ 2 = AD ^ 2 + CD ^ 2 = 196 + 196 = 2 * 196.

AC = 14 * √2 cm.

Let’s define the diagonals of the bases of the truncated pyramid.

A1C1 ^ 2 = A1D1 ^ 2 + C1D1 ^ 2 = 100 + 100 = 2 * 100.

A1C1 = 10 * √2 cm.

Diagonal section АА1С1С is an isosceles trapezoid.

Let’s build the height С1Н. The length of the segment AH = (AC + A1C1) / 2 = (14 * √2 + 10 * √2) / 2 = 12 * √2 cm.In a right-angled triangle AC1H, according to the Pythagorean theorem, C1H ^ 2 = AC1 ^ 2 – AN ^ 2 = 324 – 288 = 36.C1H = 6 cm.

Let’s determine the areas of the pyramid’s bases.

S1 = AB ^ 2 = 14 ^ 2 = 196 cm2.

S2 = A1B1 ^ 2 = 102 = 100 cm2.

Let’s define the volume of the truncated pyramid.

V = С1Н * (S1 + S2 + √S1 * S2) / 3 = 6 * (196 + 100 + √19600) / 3 = 872 cm3.

Answer: The volume of the pyramid is 872 cm3.



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