In a regular quadrangular pyramid, the height is 12 cm, and the height of the side edge is 15 cm. Find the side edge.
| February 16, 2021 | education
The height of the side face, the height of the pyramid and the OH segment form a right-angled triangle RON. Then, by the Pythagorean theorem, OH ^ 2 = PH ^ 2 – OP ^ 2 = 225 – 144 = 81.
OH = √81 = 9 cm.
Since point O is the middle of AC, point H is the middle of BC, then OH is the middle line of triangle ABC, then AB = 2 * OH = 2 * 9 = 18 cm.
Let us determine the length of the AC diagonal of the AВСD square.
AC ^ 2 = 2 * AB ^ 2 = 2 * 324 = 648.
AC = √648 = 18 * √2.
Then OС = 18 * √2 / 2 = 9 * √2 cm.
In a right-angled triangle POC, by the Pythagorean theorem, we determine the length of the hypotenuse PC.
PC ^ 2 = PO ^ 2 + OС ^ 2 = 144 + 162 = 306.
РС = √306 = 3 * √34 cm.
Answer: The length of the side is 3 * √34 cm.
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