In a regular triangular pyramid, the side of the base is 3 cm and the side edge is 2√3. find the volume.

| August 10, 2021 | education

The median CH in the equilateral triangle ABC is also its median, then AH = BH = AB / 2 = 3/2 cm.

The ACH triangle is rectangular, then by the Pythagorean theorem, CH ^ 2 = AC ^ 2 – AH ^ 2 = 9 – 9/4 = 27/4.

CH = 3 * √3 / 2 cm.

By the property of the medians, the point of their intersection divides the medians in a ratio of 2 / 1. Then CO = 2 * OH.

CH = CO + OH = 3 * √3 / 2 cm.

3 * OH = 3 * √3 / 2.

OH = √3 / 2 cm.

CO = CH – OH = 3 * √3 / 2 – √3 / 2 = √3 cm.

In a right-angled triangle COD, according to the Pythagorean theorem, DO ^ 2 = DS ^ 2 – CO ^ 2 = 12 – 3 = 9.

DO = 3 cm.

The area of ​​the base of the pyramid is equal to: Sbn = AB * CH / 2 = 3 * 3 * √3 / 4 = 9 * √3 / 4 cm2.

Vpyr = Sbn * DO / 3 = (9 * √3 / 4) * 3/3 = 9 * √3 / 4 cm3.

Answer: The volume of the pyramid is 9 * √3 / 4 cm3.



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