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Truncated Pyramid

Q: How is the height of a truncated pyramid measured?

The height, denoted by $h$, is always the perpendicular distance between the two parallel bases. It is an internal measurement from the center of the top base straight down to the plane of the bottom base.

Q: What is the difference between the height (h) and the slant height (s)?

The height ($h$) is the internal, perpendicular distance between the bases. The slant height ($s$) is the external, diagonal height of a lateral (side) face. The slant height is always longer than the perpendicular height.

Q: Does the volume formula work for a truncated pyramid with a triangular base?

Yes, the volume formula $ V = \frac{1}{3}h(A_1 + A_2 + \sqrt{A_1 A_2}) $ works for any truncated pyramid, as long as its bases are similar polygons. You would simply calculate $A_1$ and $A_2$ using the formula for the area of a triangle.

Q: Why is there a square root in the volume formula?

The $\sqrt{A_1 A_2}$ term, known as the geometric mean of the areas, arises from the algebra used to simplify the 'large pyramid minus small pyramid' calculation. It correctly accounts for the average cross-sectional area of the tapering shape.

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Ever wonder about the shape of a bucket, a lampshade, or the base of a famous monument? You're likely looking at a truncated pyramid! This lesson will guide you through this fascinating 3D shape, showing you how to calculate its volume and surface area with confidence.

Truncated Pyramid — an original Algebra911 reference diagram defining truncated pyramid with its key formula and a worked example. — Truncated Pyramid: Volume, Surface Area, and Properties

What Is a Truncated Pyramid?

A truncated pyramid is a pyramid with its top cut off by a plane that is parallel to its base. The part of the pyramid that remains between the cutting plane and the original base is the truncated pyramid. This shape is also frequently called a frustum, which is a Latin word meaning 'piece cut off'.

Think of a classic pyramid, like the ones in Egypt. Now, imagine slicing off the top section with a giant, perfectly level blade. The pointy top part is a smaller, complete pyramid, and the large bottom part you're left with is the truncated pyramid. This shape is very common in architecture and design.

A truncated pyramid has several key components:

Lower Base ( $A_{1}$ ): This is the original, larger base of the pyramid.
Upper Base ( $A_{2}$ ): This is the new, smaller top surface created by the cut. It is always geometrically similar to the lower base.
Lateral Faces: These are the side faces connecting the two bases. Unlike a full pyramid which has triangular faces, a truncated pyramid has trapezoidal faces.
Height ( $h$ ): This is the shortest, perpendicular distance between the centers of the two parallel bases.
Slant Height ( $s$ ): This is the height of one of the trapezoidal lateral faces, measured along its surface. It is always longer than the perpendicular height.

Key Properties and Terminology

Understanding the properties of a truncated pyramid is essential for working with its formulas. The relationship between its parts is predictable and follows clear geometric rules.

Parallel and Similar Bases: The most important property is that the upper and lower bases are parallel. Furthermore, the upper base is a mathematically similar, smaller version of the lower base. This means if the lower base is a square, the upper base must also be a square. If the lower base is a 6:8 rectangle, the upper base must be a rectangle with the same side ratio, like 3:4.
Trapezoidal Lateral Faces: Because the top is sliced off parallel to the bottom, the original triangular faces of the pyramid become trapezoids. In a right truncated pyramid (where the apex of the original pyramid was directly above the center of the base), these faces are identical isosceles trapezoids.
Height vs. Slant Height: It's crucial not to confuse the perpendicular height $h$ with the slant height $s$ . The height $h$ is an internal measurement used for volume calculations. The slant height $s$ is an external measurement along the slanted face, used for surface area calculations. You can often use the Pythagorean theorem with the height and parts of the bases to find the slant height if it's not given.

Here is a table comparing a full pyramid to a truncated pyramid:

Feature	Full Pyramid	Truncated Pyramid (Frustum)
Number of Bases	1	2 (an upper and a lower base)
Shape of Base(s)	Any polygon	Two similar polygons
Lateral Faces	Triangles	Trapezoids
Apex	Yes (a single point at the top)	No (has a flat top base instead)

How to Calculate the Volume of a Truncated Pyramid

The volume of a truncated pyramid is the amount of space it occupies. The formula might look complex at first, but it cleverly combines the areas of both bases with the height to give an exact result. It works for any truncated pyramid, regardless of the shape of its polygonal bases (square, triangular, hexagonal, etc.), as long as they are parallel and similar.

The formula for the volume $V$ is:

V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})

Let's break down the components:

$V$ is the volume of the truncated pyramid.
$h$ is the perpendicular height (the distance between the two bases).
$A_{1}$ is the area of the larger base (the bottom).
$A_{2}$ is the area of the smaller base (the top).

The term $\sqrt{A_{1} A_{2}}$ is the geometric mean of the two base areas. This part of the formula accounts for how the pyramid tapers from the larger base to the smaller one.

Example 1

A truncated pyramid has a height of $12$ cm. Its lower base is a square with a side length of $10$ cm, and its upper base is a square with a side length of $5$ cm. Calculate its volume.

Step 1: Calculate the area of the lower base ( $A_{1}$ ).
The base is a square with side length $10$ cm.
$A_{1} = {side}^{2} = 10^{2} = 100 {cm}^{2}$

Step 2: Calculate the area of the upper base ( $A_{2}$ ).
The base is a square with side length $5$ cm.
$A_{2} = {side}^{2} = 5^{2} = 25 {cm}^{2}$

Step 3: Substitute the values into the volume formula.
We have $h = 12$ , $A_{1} = 100$ , and $A_{2} = 25$ .
$V = \frac{1}{3} (12) (100 + 25 + \sqrt{100 \cdot 25})$

Step 4: Simplify the expression.
First, calculate the term under the square root: $\sqrt{100 \cdot 25} = \sqrt{2500} = 50$ .
Now, substitute this back into the formula:
$V = \frac{1}{3} (12) (100 + 25 + 50)$
$V = 4 (175)$

Step 5: Calculate the final volume.
$V = 700 {cm}^{3}$
The volume of the truncated pyramid is $700$ cubic centimeters.

Where Does the Volume Formula Come From?

Memorizing a formula is useful, but understanding where it comes from provides a much deeper knowledge. The volume formula for a truncated pyramid is derived from a simple, elegant idea: subtraction.

Imagine the original, complete pyramid before its top was cut off. Let's call this the 'large pyramid'. The piece that was cut off the top is also a pyramid, just a smaller version. Let's call this the 'small pyramid'.

The volume of the truncated pyramid is simply the volume of the large pyramid minus the volume of the small pyramid.

Volume (Truncated) = Volume (Large Pyramid) - Volume (Small Pyramid)

So, why don't we just use this subtraction method every time? The problem is that we usually don't know the height of the original large pyramid or the small pyramid that was removed. We are typically only given the dimensions of the frustum itself: its height $h$ , and the dimensions of its two bases.

The formula $V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})$ is the result of using algebra and the properties of similar triangles to express the volumes of the large and small pyramids in terms of the frustum's dimensions. It's a powerful shortcut that saves us from having to calculate the dimensions of the 'missing' parts of the pyramid.

How Do You Find the Surface Area of a Truncated Pyramid?

The total surface area (TSA) of a truncated pyramid is the sum of the areas of all its faces. This includes the area of the top base, the bottom base, and all the trapezoidal lateral faces.

The general formula is straightforward:

T S A = A_{1} + A_{2} + L S A

Here, $A_{1}$ and $A_{2}$ are the areas of the lower and upper bases, just like in the volume formula. The new term is $L S A$ , which stands for Lateral Surface Area. The LSA is the combined area of all the trapezoidal side faces.

If the truncated pyramid is a right pyramid (meaning the sides are all sloped at the same angle), we can use a convenient formula for the LSA:

L S A = \frac{1}{2} (P_{1} + P_{2}) s

Where:

$P_{1}$ is the perimeter of the lower base.
$P_{2}$ is the perimeter of the upper base.
$s$ is the slant height of the lateral faces.

Example 2

Let's find the total surface area of the truncated pyramid from Example 1. It has a square lower base of side $10$ cm, a square upper base of side $5$ cm, and a perpendicular height of $12$ cm. Let's assume its slant height $s$ is $12.26$ cm.

Step 1: Find the areas of the bases ( $A_{1}$ and $A_{2}$ ).
We already calculated these: $A_{1} = 100 {cm}^{2}$ and $A_{2} = 25 {cm}^{2}$ .

Step 2: Find the perimeters of the bases ( $P_{1}$ and $P_{2}$ ).
For the lower square base: $P_{1} = 4 \times 10 = 40 cm$ .
For the upper square base: $P_{2} = 4 \times 5 = 20 cm$ .

Step 3: Calculate the Lateral Surface Area (LSA).
Using the LSA formula with $s = 12.26$ cm:
$L S A = \frac{1}{2} (P_{1} + P_{2}) s$
$L S A = \frac{1}{2} (40 + 20) (12.26)$
$L S A = \frac{1}{2} (60) (12.26) = 30 \times 12.26 = 367.8 {cm}^{2}$

Step 4: Calculate the Total Surface Area (TSA).
Now, add the areas of the two bases to the LSA.
$T S A = A_{1} + A_{2} + L S A$
$T S A = 100 + 25 + 367.8$
$T S A = 492.8 {cm}^{2}$

The total surface area of the truncated pyramid is $492.8$ square centimeters.

Key formulas for truncated pyramid by Algebra911.

Worked Example: A Concrete Support Pillar

Many real-world objects use the shape of a truncated pyramid for stability and design. Let's work through a practical problem.

Example 3

A concrete support pillar is shaped like a truncated pyramid with a rectangular base. The lower base measures $4$ meters by $3$ meters. The upper base measures $2$ meters by $1.5$ meters. The pillar has a perpendicular height of $5$ meters. What is the volume of concrete needed to construct this pillar?

Step 1: Identify the given information.
Height $h = 5$ m.
Lower base dimensions: $4$ m and $3$ m.
Upper base dimensions: $2$ m and $1.5$ m.

Step 2: Calculate the area of the lower base ( $A_{1}$ ).
The base is a rectangle.
$A_{1} = length \times width = 4 \times 3 = 12 m^{2}$

Step 3: Calculate the area of the upper base ( $A_{2}$ ).
The upper base is also a rectangle. Notice that the side ratio $2 : 1.5$ is the same as $4 : 3$ , so the bases are similar.
$A_{2} = length \times width = 2 \times 1.5 = 3 m^{2}$

Step 4: Substitute the values into the volume formula.
$V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})$
$V = \frac{1}{3} (5) (12 + 3 + \sqrt{12 \cdot 3})$

Step 5: Simplify the expression.
First, the square root term: $\sqrt{12 \cdot 3} = \sqrt{36} = 6$ .
Now, substitute this back:
$V = \frac{5}{3} (12 + 3 + 6)$
$V = \frac{5}{3} (21)$

Step 6: Calculate the final volume.
$V = 5 \times 7 = 35 m^{3}$
The volume of concrete needed for the pillar is $35$ cubic meters.

Common Mistakes to Avoid

When working with truncated pyramids, a few common pitfalls can lead to incorrect answers. Being aware of them is the first step to avoiding them.

Confusing Height $h$ and Slant Height $s$ : This is the most frequent error. Remember, the volume formula always uses the perpendicular height $h$ . The lateral surface area formula uses the slant height $s$ . Don't mix them up.
Forgetting the Square Root Term in the Volume Formula: Many students accidentally calculate $V = \frac{1}{3} h (A_{1} + A_{2})$ , which is incorrect. That middle term, $\sqrt{A_{1} A_{2}}$ , is essential and cannot be omitted.
Errors in Base Area or Perimeter Calculations: Before you even start with the frustum formulas, make sure you are correctly calculating the areas (e.g., $s^{2}$ for a square, $l w$ for a rectangle) and perimeters (sum of side lengths) of the bases. A small error here will lead to a wrong final answer.
Incomplete Surface Area: When asked for the total surface area, it's easy to find the lateral surface area (LSA) and forget to add the areas of both the top and bottom bases. The total surface area must include all surfaces.
Unit Mismatch: Ensure all your measurements are in the same units before you begin calculations. If a height is in meters and a base length is in centimeters, you must convert one of them first.

Quick Summary and Key Formulas

This section provides a quick reference for the main concepts and formulas related to the truncated pyramid (frustum).

A truncated pyramid is what remains after a smaller pyramid is cut from the top of a larger pyramid with a plane parallel to the base. It features two parallel, similar bases and trapezoidal lateral faces.

Key Formulas

Volume:
Measures the 3D space inside the frustum.

V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})

Total Surface Area (TSA):
The sum of the areas of all faces.

T S A = A_{1} + A_{2} + L S A

Lateral Surface Area (LSA) for a Right Frustum:
The sum of the areas of the side faces only.

L S A = \frac{1}{2} (P_{1} + P_{2}) s

Variable Definitions

$h$ = Perpendicular height
$s$ = Slant height
$A_{1}$ = Area of the lower (large) base
$A_{2}$ = Area of the upper (small) base
$P_{1}$ = Perimeter of the lower base
$P_{2}$ = Perimeter of the upper base

Frequently Asked Questions

What is another name for a truncated pyramid?

A truncated pyramid is also commonly called a 'frustum'. The term 'frustum of a pyramid' is also used to be more specific. Frustum is a Latin word that means 'piece cut off'.

Are the side faces of a truncated pyramid always rectangles?

No, the side faces of a truncated pyramid are never rectangles. They are always trapezoids, because the top and bottom edges of each face (which are sides of the bases) are parallel but have different lengths.

How is the height of a truncated pyramid measured?

The height, denoted by $h$ , is always the perpendicular distance between the two parallel bases. It is an internal measurement from the center of the top base straight down to the plane of the bottom base.

What is the difference between the height (h) and the slant height (s)?

The height ( $h$ ) is the internal, perpendicular distance between the bases. The slant height ( $s$ ) is the external, diagonal height of a lateral (side) face. The slant height is always longer than the perpendicular height.

Does the volume formula work for a truncated pyramid with a triangular base?

Yes, the volume formula $V = \frac{1}{3} h (A_{1} + A_{2} + \sqrt{A_{1} A_{2}})$ works for any truncated pyramid, as long as its bases are similar polygons. You would simply calculate $A_{1}$ and $A_{2}$ using the formula for the area of a triangle.

Can a cone be truncated as well?

Absolutely. When you slice the top off a cone with a plane parallel to its circular base, the remaining shape is called a truncated cone, or a frustum of a cone. The formulas are similar but use radius and pi for the circular bases.

Why is there a square root in the volume formula?

The $\sqrt{A_{1} A_{2}}$ term, known as the geometric mean of the areas, arises from the algebra used to simplify the 'large pyramid minus small pyramid' calculation. It correctly accounts for the average cross-sectional area of the tapering shape.