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Finding the Volume of a Trapezoidal Prism - Made Easy With Examples

Finding the Volume of a Trapezoidal Prism (with Examples)
Stumped over calculations involving the volume of a trapezoidal prism? You need not worry, as ScienceStruck will provide you with the method to find its volume with some examples.
ScienceStruck Staff
Last Updated: Mar 26, 2018
Did You Know?
Most of the problems concerning trapezoidal prisms involve symmetrical shapes, i.e., the height on all sides is constant. But, in certain prismatic structures, the height may differ on different edges resulting in an asymmetrical trapezoidal prism.
A trapezoidal prism is a three dimensional geometric shape that consists of a trapezoid or trapezium shape on one cross section, and a rectangle on the other cross sections. The most important components of this geometric shape are its length, height, slant height, base width, and top width. With their values known, it's possible to calculate the volume and surface area of a trapezoidal prism.
The following section will give you a step-by-step explanation of calculating the volume of a trapezoidal prism and its formula. Mostly, the calculations can be done if only the bottom and top width, height, and length are known. But when the slant height is known instead of the actual height, then a different formula needs to be used.
Volume of a Trapezoidal Prism
Trapezoidal Prism
By referring to the above diagram, all the components of a trapezoidal prism are understood. Let's say that one knows the top and bottom width values, length, and height of the figure. Then the formula for calculating its volume is:
Volume (V) = L x H x ((P+Q)/2) ---- (equation 1)

where,

L - Length

H - Height

P - Top width

Q - Bottom width
In case the slant height is given instead of the actual height, then the formula to calculate the volume is:
Volume (V) = L x (P+Q) x √(4S2+ 2PQ - Q2- P2)/4 ---- (equation 2)

where S - Slant height
Other values have the same names as mentioned in the first formula. The √ sign indicates square root of the entire bracket value.

Go through the following examples to understand how the volume of a trapezoidal prism is calculated.
Examples
Example #1
Calculate the volume of a trapezoidal prism having a length of 7 centimeters and a height of 4 centimeters. The top and bottom widths are 3 and 2 centimeters respectively.

Solution
The given data consists of:

H = 4 cm

L = 7 cm.

P = 3 cm.

Q = 2 cm.
Thus, by using equation no. 1, i.e., the first formula, the expression can be written as:
Volume (V) = 7 x 4 x ((3+2)/2)
= 28 x 2.5

= 70
Thus, the volume of the prism is 70 cubic centimeters (cc).
Example #2
A trapezoidal prism has a length of 5 cm and bottom width of 11 cm. The top width is 6 cm, and slant height is 2 cm. Find the volume of this geometric structure.

Solution
The given data consists of:

S = 7 cm

L = 5 cm.

P = 2 cm.

Q = 6 cm.
As the actual height is not given, we have to use equation no. 2 for solving this problem. The expression can be written as:
Volume (V) = 5 x (2+6) x √(4 x (72) + 2(2 x 6) - 62 - 22)/4

V = 40 x (√(196 + 24 - 36 - 4)/4)

= 40 x √(180/4)

= 40 x 6.70

= 268
Thus, the volume of the prism is 268 cubic centimeters (cc).
Always remember to use the right units when you find the volume, as sometimes instead of centimeters, even inches and millimeters can be used for expressing the given data. Also, in case of any problem where all the values of the trapezoidal prism are given in different units, remember to convert them to a unit that you are comfortable with before proceeding with the calculations.