The Volume of a Trapezoid: Meaning and Formula

The Volume of a Trapezoid: Introduction

A trapezoid or trapezium is a two-dimensional figure( quadrilateral). One pair of sides of the trapezoid is parallel to each other, called the base(s) of the trapezoid. The other pair of sides are non-parallel sides are termed as the legs of the trapezoid. The distance between the bases of the trapezoid is called altitude or height. For any quadrilateral, we can calculate area and perimeter. In this article, we will learn to find out the volume of a three-dimensional trapezium. The Volume of a trapezoid is the space occupied by the object in cubic units.

Formula to calculate the volume

The formula for finding the area of a 2D trapezoid is A=\frac{1}{2}\times h\times (b_1+b_2).

Here A is the area of a trapezoid.

h is the perpendicular distance between the bases of the trapezoid(height).

b_1\text{ and }b_2 are the lengths of the bases of the trapezoid.

But we cannot find the volume of a two-dimensional figure. To find out the volume of a trapezoid, you have to construct a 3D figure with two trapezoidal bases. We can find the volume of the 3D trapezoid simply by multiplying the area of the trapezoidal base with the length of the 3D figure.

Let us consider:

l = length of the 3D figure.

b\text{ and }B = length of the parallel sides of the trapezoidal base.

h = distance between the parallel sides of the trapezoidal base(height).

Formula to find the volume of 3D trapezoid uses four variables,

V=\frac{1}{2}\times l\times h\times (b+B)

We can rewrite the formula as

V = Area of base × length of 3D figure

Since the area of the trapezoid =\frac{1}{2}\times h\times (b+B)

Examples with Solution

Example 1:

Find the volume of a trapezoidal prism given that the length of the parallel sides are 10 cm and 8 cm respectively, the length of the prism is 4 cm, and the height of the trapezoidal base is 6 cm.

Solution:

A trapezoidal prism is a polyhedron with two faces in the shape of a trapezoid. The side faces of the 3D object are rectangles. The trapezoidal faces are congruent to each other. There are 6 faces (2 trapezoidal and 4 rectangular) and 8 vertices. 

In the given question, 

The base 1 = 10 cm, base 2 = 8 cm, the height or perpendicular distance between the parallel sides of the trapezoidal base is 6 cm, and the length of the prism is 4 cm. 

Area of trapezium/trapezoid

=\frac{1}{2}\times h\times (b_1+b_2)

=\frac{1}{2}\times 6\times (10+8)=54\text{ } cm^2

The volume of the trapezoid prism

    V = Area of base × length of the prism.

⇒ V = 54\text{ } cm^2\times 4\text{ } cm

        = 216\text{ } cm^3

Answer: Thus, the Volume of the trapezoid prism is 216 cm^3.

Example 2:

Find the length of the trapezoidal tank, when the area of the base is 60 m^2 and the tank capacity is 360 m^3?

Solution:

Given:

The Volume of the tank = 360 m^3.

Area of the base = 60 m^2

The Volume of a trapezoid tank V= Area of base ×length of tank.

360\text{ }m^3 = 60\text{ }m^2\times l

⇒ l=\frac{360\text{ }m^3}{60\text{ }m^2}

∴ l=6\text{ }m

Answer: The length of the trapezoid tank is 6 m.

Practice Multiple Choice Questions