The sides of the base of a regular hexagonal pyramid are 14, the side edges are 25.

The sides of the base of a regular hexagonal pyramid are 14, the side edges are 25. Find the area of the lateral surface of this pyramid.

Regular hexagonal pyramid SABCDEF.
The lateral surface area of ​​a regular pyramid is calculated by the formula:
S = P * h / 2,
where P is the perimeter of the base, h is the apothem.
Apothem is the height of the side face. Since the pyramid is correct by condition, all of its lateral faces are equal and represent isosceles triangles with lateral sides, which are the lateral edges of the pyramid, and a base, which is the side of the pyramid base. Consider one of the lateral faces of SAB: SA = SB = 25, AB = 14, SH is the height of the triangle (apothem of the lateral edge). Since SH is the height drawn to the base, then SH and the median, thus: AH = BH = AB / 2 = 14/2 = 7.
Along the tower of Pythagoras:
SH = √ (SA ^ 2 – AH ^ 2) = √ (25 ^ 2 – 7 ^ 2) = √ (625 – 49) = √576 = 24.
The perimeter of the base is:
P = 6AB = 6 * 14 = 84.
Then:
S = 84 * 24/2 = 2016/2 = 1008.
Answer: S = 1008.




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