A pyramid with different altitudes. Solve for volume.?

Question

A pyramid has a rectangular base 10 cm by 25 cm and four triangular faces; two opposing faces both with altitude 13 cm and the other two faces with altitudes of 20 cm and 15 cm. What is the volume of the pyramid, in cubic cm?

It's for Math Counts. Please do not solve it for me, because I want to solve myself. Instead, explain the steps to me, such as using examples.

Please list the steps too.
Such as 1) Add all altitudes and find average, etc...

I will choose a best answer if it is the one that helps me the most. Thank you~

Answer 1

You need to find the altitude of the pyramid, using the altitudes of the sides.

The altitudes of 13 cm will be from the long sides of the pyramid to the apex. This will give a cross-section of the pyramid perpendicular to the long sides of the base of an equilateral triangle with base 10 and sides 13. Find the altitude of this triangle.

The cross-section perpendicular to this will have a base of 25 with sides of 15 and 20. Find the altitude of this triangle If the altitudes of these two triangles are the same, you may proceed. P.S. - They are the same.

The altitudes of these two triangles are the true altitude that you need to plug into the pyramid volume formula along with the base area, which should be simple enough to calculate, and the mid-height area, also simple to calculate because the dimensions will each be one-half the base dimensions.

If that formula has not been given to you yet, contact me through here and I'll give it to you.

Answer 2

The vertex (top) of the pyramid is off center. Draw the view from the top. Put the point for the vertex equal distances from the 2 long sides but closer to one of the short sides than the other. Draw solid lines for the edges, dotted lines for the altitudes of the faces. Focus on one of those 13 cm altitudes. It's the hypotenuse of a right triangle, one side (hiding under it in the diagram) is half the 10 cm short side of the base, 5 cm. The other side is the altitude you want, hiding vertically under the vertex of the pyramid. So you're looking at a 5 - x - 13 right triangle, and of course x must be 12. So the altitude is 12, and volume is (1/3)(12)(250) = 1000 cm³.

As a double check, under the 20 cm face altitude should be an x - 12 - 20 right triangle, which makes x 16, and under the 15 cm face altitude should be an x - 12 - 15 right triangle, which makes x 9, and sure enough the 2 x's, 16 and 9, add up to the 25 cm length of the base.