Volume of a Truncated Pyramid
The volume of a truncated pyramid is the capacity of a truncated pyramid. A pyramid is a polyhedron, that is a 3D solid figure with a polygonal base and a top point called an apex. When a pyramid is sliced along its crosssection, a smaller pyramid and a truncated pyramid are obtained. In this section, we will learn about the volume of a truncated pyramid. Stay tuned to learn more!!!
What is Volume of a Truncated Pyramid?
The volume of a truncated pyramid is the number of cubic units that can be held by a truncated pyramid. A pyramid has an apex and only one base face whereas a truncated pyramid does not have an apex and has two base faces, one at the top and one at the bottom. Only the base of a pyramid is a polygon, the rest of the faces are triangles. A pyramid may be a 'right' in which its apex is directly over the centroid over its base or else a pyramid can be 'oblique' which are basically nonright pyramids. Pyramids are named after their bases, for example, a pyramid with a triangle base is called a triangular base, a pyramid with a square base is called a square pyramid, a pyramid with an octagonal base is called an octagonal pyramid, and so on. A pyramid with an 'n' sided base has 'n+1' vertices, 'n+1' faces, and '2n' edges. In the case of a truncated pyramid, both the base faces must have equal sides, therefore, a truncated pyramid with 'n' sided base faces has '2n' vertices, 'n+2' faces, and '3n' edges. For example, it can be expressed as m^{3}, cm^{3}, in^{3}, etc depending upon the given units.
Formula of Volume of a Truncated Pyramid
Since we know that a truncated pyramid is formed by slicing off a pyramid along its crosssection, this concept will help us understand the formula volume of a truncated pyramid.
Therefore, the volume of a truncated pyramid = Volume of the whole pyramid  Volume of the small pyramid.
Thus, the formula of volume of a truncated pyramid is V = 1/3 × h × (a^{2} + b^{2} + ab) where "V", "h", "a" and "b" are volume of the truncated pyramid, height of the truncated pyramid, the side length of the base of the whole pyramid, and the side length of the base of the smaller pyramid.
Derivation of Volume of a Truncated Pyramid
Let's now find the formula of the volume of a truncated pyramid.
Let us consider that the base of the whole pyramid is a square of side length "a" units and the base of the small pyramid at the top is a square of side length "b" units. Also, let us consider the height of the whole pyramid as "H" units, the height of the truncated pyramid to be "h" units, therefore, the height of the small pyramid will be "Hh" units.
Volume of a truncated pyramid, V = Volume of the whole pyramid  Volume of the small pyramid.
⇒ V = (1/3 × Base area of the whole pyramid × Height of the whole pyramid)  (1/3 × Base area of the small pyramid × Height of the small pyramid)
⇒ V = (1/3 × a^{2} × H)  (1/3 × b^{2} × (H  h)
⇒ V = 1/3 × (a^{2}H  b^{2}(H  h))
⇒ V = 1/3 × (a^{2}H  b^{2}H + b^{2}h)
⇒ V = 1/3 × ((a^{2}  b^{2})H + b^{2}h) ...(1)
Now, the ratio between the heights of the whole pyramid and the small pyramid, H:(H  h) = a:b. Therefore, the ratio between the whole pyramid and the truncated pyramid would be, H: h = a:(a  b)
⇒ H/h = a/(a  b)
⇒ H = (a/(a  b)) × h ...(2)
Substituting the value of "H" from equation (2) in the equation (1), we get:
⇒ V = 1/3 × ((a^{2}  b^{2})(a/(a  b)) × h) + b^{2}h))
⇒ V = 1/3 × (((a + b)(a  b) × a)/(a  b)) × h + b^{2}h
⇒ V = 1/3 × (a + b) × a × h + b^{2}h
⇒ V = 1/3 × h × (a^{2} + b^{2} + ab)
Thus, the volume of a truncated pyramid is given as, V = 1/3 × h × (a^{2} + b^{2} + ab).
How to Find the Volume of a Truncated Pyramid?
We can find the volume of a truncated pyramid using the following steps:
 Step 1: Identify the given dimensions as the height of the truncated pyramid, the side length of the base of the whole pyramid, and the side length of the smaller pyramid.
 Step 2: Substitute the values in the formula V = 1/3 × h × (a^{2} + b^{2} + ab) to determine the value of the volume of a truncated pyramid.
 Step 3: Write the obtained answer with cubic units.
Example: Find the volume of a truncated pyramid whose height is 9 units if the side length of the base of the whole pyramid is 5 units and the side length of the base of the smaller pyramid is 3 units.
Solution: Given the h = 9 units, a = 5 units and b = 3 units.
Volume of truncated pyramid, V = 1/3 × h × (a^{2} + b^{2} + ab)
⇒ V = 1/3 × 9 × (5^{2} + 3^{2} + 15)
⇒ V = 3 × 49 = 147 cubic units
Thus, the volume of the truncated pyramid is 147 cubic units.
Solved Examples on Volume of a Truncated Pyramid

Example 1: Find the volume of a truncated square pyramid whose height is 12 cm and the side length of the top face is 3 cm and the side length of the bottom face is 4 cm.
Solution: Given, height of the truncated square pyramid = 12 cm, side length of the top face, b = 3 cm and side length of the bottom face, a = 4 cm
Therefore, the volume of the truncated square pyramid, V = 1/3 × h × (a^{2} + b^{2} + ab)
⇒ V = 1/3 × 12 × (4^{2} + 3^{2} + 4 × 3)
⇒ V = 4 × (16 + 9 + 12)
⇒ V = 4 × 37
⇒ V = 148 cm^{3}Answer: The volume of a truncated square pyramid is 148 cm^{3}.

Example 2: If the volume of a whole pyramid is 480 cm^{3}. The ratio between the volume of the whole pyramid and the small pyramid is 8:3, find the volume of the truncated pyramid.
Solution: Given, the volume of a whole pyramid = 480 cm^{3}
The ratio between the volume of the whole pyramid and the small pyramid = 8:3From the ratio, let the volume of the small pyramid be '3x', therefore, the volume of the whole pyramid will be '8x'
Therefore, 8x = 480
⇒ x = 480/8 = 60The volume of the small pyramid will be = 3x = 3 × 60 = 180 cm^{3}
Therefore, the volume of the truncated pyramid = Volume of the whole pyramid  Volume of the small pyramid
⇒Volume of the truncated pyramid = 480  180 = 300 cm^{3}
Thus, the volume of the truncated pyramid is 300 cubic centimeters.
FAQs on the Volume of a Truncated Pyramid
What is the Volume of the Truncated Pyramid?
The volume of a truncated pyramid is defined as the capacity of a truncated pyramid. A truncated pyramid is obtained when we slice off a pyramid along its crosssection. Thus, the volume of the truncated pyramid is obtained when the volume of the smaller pyramid is subtracted from the volume of the whole pyramid.
What is the Formula of the Volume of the Truncated Pyramid?
The volume of a truncated pyramid is given by the formula, V = 1/3 × h × (a^{2} + b^{2} + ab) where "V", "h", "a" and "b" are volume of the truncated pyramid, height of the truncated pyramid, the side length of the base of the whole pyramid, and the side length of the base of the smaller pyramid.
What Are the Units Used When You Find the Volume of a Truncated Pyramid?
The unit of volume of a truncated pyramid is "cubic units". For example, it can be expressed as m^{3}, cm^{3}, in^{3}, etc depending upon the given units.
How Can We Find the Volume of a Truncated Pyramid?
We can find the volume of a truncated pyramid using the below steps:
 Step 1: Identify the given dimensions as the "h", "a", and "b".
 Step 2: Now determine the value of the volume of a truncated pyramid by substituting the values in the formula V = 1/3 × h × (a^{2} + b^{2} + ab).
 Step 3: Represent the obtained answer with cubic units.
How to Find the Volume of a Truncated Pyramid If the Volumes of the Whole Pyramid and the Smaller Pyramid are Given?
We can find the volume of a truncated pyramid if the volumes of the whole pyramid and smaller pyramid are given using the below steps:
 Step 1: Identify the given dimensions as the volumes of the whole pyramid and the smaller pyramid.
 Step 2: Now determine the value of the volume of a truncated pyramid by substituting the values in the formula V = Volume of the whole pyramid  Volume of the small pyramid.
 Step 3: Represent the obtained answer with cubic units.
What Happens to the Volume of a Truncated Pyramid If the Height of the Pyramid is Doubled?
The volume of a truncated pyramid is doubled if the height of the pyramid is doubled as "h" is substituted as "2h", thereby giving the formula V = 1/3 × h × (a^{2} + b^{2} + ab) = 1/3 × (2h) × (a^{2} + b^{2} + ab) = 2(1/3 × h × (a^{2} + b^{2} + ab)) which gives twice the original value of volume.
What Happens to the Volume of a Truncated Pyramid If the Height of the Pyramid is Halved?
The volume of a truncated pyramid is halved if the height of the pyramid is halved as "h" is substituted as "h/2", thereby giving the formula V = 1/3 × h × (a^{2} + b^{2} + ab) = 1/3 × (h/2) × (a^{2} + b^{2} + ab) = (1/2) × (1/3 × h × (a^{2}+ b^{2} + ab)) which gives half the original value of volume.