SOLUTION: A right pyramid has a square base and a lateral area of 144 in2. If the slant height is twice the length of a base edge, find the volume.

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Question 567893: A right pyramid has a square base and a lateral area of 144 in2. If the slant height is twice the length of a base edge, find the volume.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
right pyramid with square base and lateral area of 144 square inches.
slant height is twice the length of a base edge.
what is the volume?

the faces of your pyramid are the base plus 4 isosceles triangles.
the 4 isosceles triangle form the lateral area.
divide 144 by 4 and you get the area of 1 of the isosceles triangles.
the slant height of the pyramid is equal to the perpendicular bisector of that isosceles triangle.
this also makes it the height of that isosceles triangle.
the base is set to x.
the slant height is set to 2x.
the area of the isosceles triangle is 36 square inches.
the formula for the area of a triangle is bh/2 = A which can also be written as bh = 2A
since A of the triangle is 36, this formula becomes bh = 72
since b = x and h = 2x, this formula becomes 2x^2 = 72
divide both sides of this equation by 2 to get x^2 = 36
take the square root of both sides of this equation to get x = 6.
this makes the height of the triangle equal to 2x which is equal to 12.
since the height of the triangle is 12, this makes the slant height of the pyramid also equal to 12 because the altitude of one of the faces of the pyramid is the slant height of the pyramid.
you can now find the height of the pyramid by extending a line from the foot of one of the heights of one face of the pyramid to the foot of the height of the opposite face of the pyramid.
this forms another isosceles triangle with the equal sides being 12 and the base being 6.
you can now find the height of the pyramid which will be equal to sqrt(135)
you can now find the volume of the pyramid because the volume of the pyramid equals b^2h/3.
b is equal to 36*sqrt(135)/3 which is equal to 12*sqrt(135).
the decimal equivalent would be 139.4274005
check the diagram to see what i'm talking about.
in that diagram, EG is one of the slant heights of the pyramid and is also the height of the triangular face formed by DEC.
if you slice across the middle of the base of the pyramid, you form another triangle called EHG which has a base of 6 and equal legs of 12. these equal legs are EG and EH. since EG equals 12, then EH equals 12 also.
use pythagorean formula to find EF which is the altitude of the pyramid.
once you find that, you have the volume of the pyramid because that is equal to 1/3 * the height of the pyramid times the area of the base which winds up being 1/3 * sqrt(135) * 36.