In this unit you will need a box shaped something prefer the one because that a bar of soap shown above. The mathematical name for the shape is a truncated square pyramid. 2 of its deals with are non-congruent squares, and also four are congruent trapezoids.) the is a part of a pyramid, v its top cut off. Can you find its volume?

In the unit titled 3 congruent pyramids the make a cube (in the food pack), we learned the the volume that a ideal pyramid is one 3rd times the area of its base times its height, or V = 1/3*area that base*height. The formula because that a best pyramid’s volume likewise holds because that the volume of any type of pyramid, as you deserve to see in this diagram:


This reality is also clearly shown here. Can we use this formula, V = 1/3*area the base*height, to help us uncover the volume of our soap box, a truncated pyramid? What carry out we need to do? watch the photo below!


us will need to use the formula twice, once to uncover the volume of the large pyramid, let’s speak to it BigV, and then to discover the volume of the little pyramid on top, let’s call it LittleV, and then come subtract, BigV – LittleV, i m sorry is the volume we want.


but given our soap box, how do we discover the heights that the totality pyramid and also the small pyramid? us will do it using similar triangles. Top top the diagram below a triangular challenge of the entirety pyramid is highlighted in blue, and also a triangle inside the pyramid which consists of the pyramid’s height is emphasize in yellow. Keep in mind that the blue and yellow triangles are not congruent, and the blue triangular face, i m sorry is slanted, is slightly taller 보다 the yellow triangle with the elevation inside.


The blue triangle is a challenge of the pyramid, and the yellow triangle has the pyramid’s height.We place the 2 triangles side by side on a plane:


Our goal is to discover the heights AF’ and AG’ of the two yellow triangles, AB’C’ and AD’E’. If we uncover them, we can discover the volumes of the 2 pyramids, and, by subtraction, the volume the the truncated pyramid.

notice that segment DE is the same size as segment D’E’ (each is the length of the base of the little pyramid), and segment BC is the same length as B’C’ (each is the size of the basic of the big pyramid). And with a leader we have the right to measure every the red segments in the drawing below. (F’G’ is the elevation of the truncated pyramid. We can measure it by putting an index card on height of our truncated pyramid and measuring the size from the map to the table-top.) So we are all set to collection up a proportion to find the heights AF’ and AG’.


We have actually

(little-h + F’G’) / B’C’ = little-h / D’E’, i beg your pardon is the exact same as

(little-h + F’G’) / BC = little-h / DE.

The just variable that us don’t understand in this equation is little-h. So we fix for it.

with a tiny algebra we have

little-h = (F’G’ * DE) / (BC - DE), or

little-h = (height of truncated pyramid * side size of top) / (side size of base – side size of top)

and big-h = little-h + height of truncated pyramid

So us now have actually heights that both pyramids. Us will use them in our formulas for volume.

volume of huge pyramid = 1/3 * area of base * height

= 1/3 (BC)2 * (little-h + F’G’)

volume of big pyramid =___________ cubic cm.

volume of small pyramid = 1/3 * area of base * elevation

= 1/3 (DE)2*(little-h)

volume of little pyramid = ___________ cubic cm.

volume that truncated pyramid = volume of large pyramid – volume of small pyramid

volume of truncated pyramid = __________________ cubic cm.

currently fill your truncated pyramid v rice and pour that in a graduated cylinder. Exactly how well does it match your mathematically obtained answer?

The formula because that the volume the a truncated square pyramid with elevation h, and also top leaf a cm and bottom edge b cm is V = 1/3*(a2 + abdominal + b2)*h.

You are watching: Volume of a truncated rectangular pyramid

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(The source of this formula is provided below!)

The ancient Egyptians knew the formula; view Volume the frustum that square pyramid.