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find the total area the regular pyramid.

54√3 +18√91 units²

Step-by-step explanation:

The regular hexagon base is comprised of 6 equilateral triangles, each having a side length of 6. That means the internal right triangle shown having a face edge as its hypotenuse has side lengths of 6 and 8. The face edge length is then ...

   face edge = √(6² +8²) = 10 . . . . . double the hypotenuse of a 3-4-5 right triangle, as expected

So, each of the 6 base triangles is an equilateral triangle with side length 6, and each of the 6 face triangles is an isosceles triangle with side lengths 10 and base length 6.

In each case, the altitude of the triangle can be figured from the Pythagorean theorem:

  face slant height = √(10² -3²) = √91

  base apothem = √(6² -3²) = 3√3

The area of each of these triangle is half the product of its base width and height. The total area of the pyramid is 6 times the sum of the areas of these two triangles.

  total surface area = 6·(1/2·6·√91 +1/2·6·3√3) = 18(√91 +3√3)

  total surface area = 18√91 +54√3 . . . units²

answer: 177/8

step-by-step explanation: