In a regular hexagonal prism ABCDA1B1C1D1 E1 F1 the radius of the circle inscribed in the base is 12

| July 1, 2021 | education

In a regular hexagonal prism ABCDA1B1C1D1 E1 F1 the radius of the circle inscribed in the base is 12 and the length of the side edge is 7. Find the distance between the vertices A and C1.

The radius of a circle inscribed in a regular hexagon is: R = a * √3 / 2, where a is the length of the side of the hexagon, then a = AB = 2 * R / √3 = 2 * 12 / √3 = 8 * √3 cm.

In an isosceles triangle ABC, the angle ABC is 120, then, according to the cosine theorem:

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120 = (8 * √3) ^ 2 + (8 * √3) ^ 2 – 2 * 8 * √3 * 8 * √3 * (-1/2) = 384 + 192 = 576.

AC = 24 cm

In a right-angled triangle ACC1, according to the Pythagorean theorem, we determine the length of the hypotenuse AC1.

AC1 ^ 2 = AC ^ 2 + CC1 ^ 2 = 576 + 49 = 625.

AC1 = 25 cm.

Answer: From point A to point C1 25 cm.



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