Question 1163740: The lateral edges of a regular hexagonal pyramid are all 20 cm long, and the base edgesare all 16 cm long. To the nearest cc, what is the volume of this pyramid? To the nearestsquare cm, what is the combined area of the base and six lateral faces?
Answer by greenestamps(13203)   (Show Source):
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Let A be one of the corners of the base, with edges of length 16cm.
Let P be the center of the base. Since it is a regular hexagon, AP is also 16cm.
Let V be the peak (vertex) of the pyramid. Then APV is a right triangle which has AP=16 as one leg and AV=20 as the hypotenuse.
The Pythagorean Theorem (or recognition of a 3-4-5 right triangle) gives the height as 12cm.
The volume is one-third the area of the base, multiplied by the height; use the formula for the area of a regular hexagon to find the area of the base.
For the lateral surface area, each face is an isosceles triangle with base 16. To find the slant height of each triangular face, form a right triangle with the slant height as the hypotenuse; the legs are the height of the pyramid and a segment from the center of the base to the midpoint of a side of the base.
Note the length of the segment from the center of the base to the midpoint of one side of the base is the length of the altitude of an equilateral triangle with side length 16.
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