In a regular quadrangular pyramid, the height is 5, the side edge is 13. find its volume.

| May 9, 2021 | education

The height of a regular quadrangular pyramid connects the top of the pyramid and the intersection of the diagonals (which divides them in half). The triangle formed by the height, the side edge of the pyramid and the half of the diagonal is rectangular, with the side edge being the hypotenuse. Using the Pythagorean theorem, we find the length of half of the diagonal of the square lying at the base of the pyramid:

√ (13 ^ 2 – 5 ^ 2) = √ (169 – 25) = √144 = 12.

Two adjacent halves of the diagonals and the side of the square lying at the base of the pyramid also form a right-angled triangle, with the side of the square being the hypotenuse. Let’s find the length of the side of the square using the Pythagorean theorem:

√ (12 ^ 2 + 12 ^ 2) = √ (144 * 2) = 12√2.

The volume of a regular quadrangular pyramid is equal to one third of the product of the square of the side of a regular quadrangle lying at the base by the height of the pyramid:

1/3 * (12√2) ^ 2 * 5 = 1/3 * 144 * 2 * 5 = 48 * 10 = 480.

Answer: 480.



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