A right pyramid with a square base has a total surface area of 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid?

Question

anonymous · Answer

The volume of a pyramid is 1/3*base area*height.

So we need to calculate the base area and the height of the triangle.

anonymous · Answer

To calculate the base area. We are given that the total area is 432 square units. This includes the the area of the four sides and the area of square base.

We are also given that the each of the triangular sides is equal to 1/2 of the square base area. 

So lets let s=the area of the square base.

Since we have four triangular sides and one square base, we write our equation as this:

1/2s+1/2s+1/2s+1/2s+s=432

combine like terms 

3s=432
s=144

So the base area is 144 square units.

Each of the triangular sides is 1/2*144 or 72 square units.

We still need to find the height of the pyramid

anonymous · Answer

To find the height of the pyramid, let's drop a vertical line down from the top of the pyramid to the middle of the square base.

Next draw a line from the middle of the square base to the middle of the base of one of the triangular sides. 

Next draw a line from the middle of the base of this triangular side to the top of the pyramid, called the slant length.

We now have a right triangle and we can use the Pythagorean theorem to find the height of the pyramid. 

To use the Pythagorean theorem, we need to find the slant length and the distance from the center of the square base to the middle of the triangular side.

Since the base is a square of 144 sq. units, the length of each of it's sides is the square root of 144 which is 12.

So each of the fours sides of the square base is 12 units long.  The distance from the center of the square base to one of the sides is 1/2*12 or 6 units long.

The area of one of the triangle sides is 1/2*base length * slant height.

We know the area is 72 square units and the base length is just one side of the square base or 12 units.

So we have this equation:

72=1/2*12*slant height

simplifying

72=6*slant height

rearranging

72/6= slant height

12= slant height

anonymous · Answer

We can now find the height of the pyramid using the Pythagorean theorem. 

12^2= 6^2 +height^2

simplifying

144=36 + height^2

rearranging

144-36=height^2

10.39= height

So the volume is:

1/3* 10.39*144=499 cubic units