Find the length of the diagonal of a regular quadrangular prism with a base side of 10cm and a side diagonal of 18cm.
At the base of a regular quadrangular prism lies a square, and the side faces are equal rectangles.
Consider a right-angled triangle in which the side of the base and the side edge are legs, the diagonal of the side edge is the hypotenuse. By the Pythagorean theorem, the square of the side edge can be found as the difference between the squares of the diagonal of the side face and the side of the base:
h ^ 2 = 18 ^ 2 – 10 ^ 2 = 324 – 100 = 224.
The square of the diagonal of the base is equal to the sum of the squares of the two sides of the base:
d ^ 2 = 10 ^ 2 + 10 ^ 2 = 100 + 100 = 200.
From a right-angled triangle formed by the diagonal of the prism, the side edge and the diagonal of the base, we can find the square of the diagonal of the prism:
D ^ 2 = d ^ 2 + h ^ 2 = 200 + 224 = 424;
D = √424 ≈ 20.59 cm – the diagonal of this regular quadrangular prism.
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