The word truncated means to shorten something and now you can guess what truncated pyramid means. If you think the truncated pyramid is the same as the normal pyramid then you are wrong. They both are two different shapes and deserves their own resource. We already have discussed the normal pyramids before, therefore, in this resource, we will discuss truncated pyramids.

What is a Truncated Pyramid?

At this point, you should know what is a pyramid, if not then we have a separate resource for the pyramid and you should first check it out. Imagine a pyramid, the point that makes it a perfect shape is the apex. We know it looks satisfying but we want you to cut off the apex and then, the pyramid will look something like below:

Yes, it will be called a pyramid but this isn't some regular pyramids that you came across. This one is special and mathematicians gave it a beautiful name which is, "truncated pyramid". A truncated pyramid is the result of cutting a pyramid by a plane parallel to the base and separating the part containing the apex. The lateral faces of a truncated pyramid are trapezoids. The height of a truncated pyramid is the perpendicular distance between the bases and the apothem is the height of any of its sides.

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Let's go

Unfold of a Truncated Pyramid

Let's come to the unfolding part, which is one the most demanding part. Unopened three-dimensional shapes reveal many hidden secrets that can be used for different calculations. We have already unfolded this figure and this is what we found:

Calculation of the Apothem of a Truncated Pyramid

To calculate the apothem of a truncated pyramid, the height, the apothem of the biggest base and the apothem of the minor base must be known. Apply the Pythagorean theorem to determine the length of the hypotenuse of the red triangle to obtain the apothem:

{ Ap }^{ 2 } = { h }^{ 2 } + { ({ ap }_{ 2 } - { ap }_{ 1 }) }^{ 2 }

Ap = \sqrt { { h }^{ 2 } + { ({ ap }_{ 2 } - { ap }_{ 1 }) }^{ 2 } }

Area and Volume of a Truncated Pyramid

Finding the area and volume of a truncated pyramid might be tricky because of different parts having different shapes, i.e. the polygon base and the parallelogram faces.

In order to find the area of a truncated pyramid, you need to find the parameter of both bases, the top one as well as the bottom one. Finding parameters of both bases isn't hard, all you need to do is to add all the outer sides. Usually, the letter "P" represents the parameter but since we are talking about two parameters, that can create confusion, therefore, we have made some changes in the letter.

The lateral area of a truncated pyramid is half of the sum of both parameters and then multiplied by the apothem of the pyramid.

{ A }_{ L } = \frac { P + P' }{ 2 } \times Ap

However, this isn't the total area, lateral area means the sum of face areas. We are still missing both base areas. You might have guessed it by now, we will find both base areas and then add them with the lateral area to get the total area.

{ A }_{ T } = { A }_{ L } + A + A' or { A }_{ T } = \frac { P + P' }{ 2 } \times Ap + A + A'

Where,

Finding the volume of a truncated pyramid includes the area of both bases. Furthermore, you will also need the height of the truncated pyramid. If you have all information available, use the below formula to calculate the volume of a truncated pyramid:

V = \frac { h }{ 3 } (A + A' + \sqrt { A \times A' } )

Example

Calculate the lateral area, surface area and volume of the truncated square pyramid whose larger base edge is 24 cm, smaller base edge is 14 cm and whose lateral edge is 13 cm.

Ap = 13 cm

{ Ap }^{ 2 } = { h }^{ 2 } + { 5 }^{ 2 }

{ 13 }^{ 2 } = { h }^{ 2 } + { 5 }^{ 2 }

h = \sqrt { { 13 }^{ 2 } - { 5 }^{ 2 } } = 12 cm

A = { 24 }^{ 2 } = 576 { cm }^{ 2 }

A' = { 14 }^{ 2 } = 196 { cm }^{ 2 }

{ A }_{ T } = 912 + 576 + 196 = 1684 { cm }^{ 2 }

V = \frac { 12 }{ 3 } \times (576 + 196 + \sqrt { 576 \times 196 }) = 4432 { cm }^{ 3 }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.