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Basic Geometry: Help & Review16 chapters | 109 lessons

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Lesson Transcript

Instructor:
*David Karsner*

Trapezoidal prisms are three dimensional solids. They consist of two congruent trapezoids that are connected to each other by four rectangles. This lesson will include the definition and how to find the surface area and the volume of these solids.

The term trapezoidal prism may not familiar to you. If you've ever noticed the shape of most circular fire-pit bricks, then you will be familiar with trapezoidal prisms. They look like a typical brick except one side is shorter than the other. This lesson is about describing a trapezoid and a trapezoidal prism. It will also cover the formulas for the surface area and volume of these prisms.

Before we can talk about what a trapezoidal prism is, we need to know what a trapezoid is. **Trapezoids** are quadrilaterals (four sided figures) that have only one set of parallel lines. They are different from other types of quadrilaterals (parallelograms, rectangles, squares) that have two sets of parallel lines. The two sides of the trapezoid that are parallel are called bases, and they won't be the same length. To find the surface area and volume of a trapezoidal prism, you will need to know the formula for finding the area of a trapezoid. The area of a trapezoid is *h*(*b*1+*b*2)/2, where *b1* and *b2* are the lengths of the two bases and *h* is the height of the trapezoid.

A **trapezoidal prism** is a three dimensional solid that has two congruent trapezoids for its top and lower base. It will have four rectangles that connect the corresponding sides of the two bases. These four rectangles are called the lateral faces of the prism.

To find the surface area, the amount of space on the outside of a solid, of a trapezoidal prism, you'll need to find the area of the trapezoid base and multiply it by two. You'll then add the area of the four rectangles to the area of the two bases. The width of the four rectangles will be the same as the height of the prism. The length of the four rectangles will be the sum of the four sides of the trapezoid, the two bases, and the other two sides. The surface area can be given with this formula: Surface Area of Trapezoidal Prism = (*b*1+*b*2)*h* + *PH*. In this formula, *b*1 and *b*2 stand for the length of the bases of the trapezoid. Lower case *h* is the height of the trapezoid. Upper case *P* is the perimeter of the trapezoid, and upper case *H* is the height of the prism.

To find the volume, the amount of space on the inside of a solid, you will need to find the area of the trapezoid base and then multiply it by the height of the prism. Keep in mind that there are two different heights. There is the height of the trapezoid, which you will use to find the area of the trapezoid. There is also the height of the prism, which you will need to find the volume of the trapezoidal prism. The formula for the volume of trapezoidal prism is volume=*Hh*(*b*1+*b*2)/2. In this formula, *b*1 and *b*2 are the base of the trapezoid. Lower case *h* is the height of the trapezoid, and upper case *H* is the height of the prism.

Here's our first example:

Find the surface area of a trapezoidal prism that has bases of 3 and 5 feet, sides of 3 feet, a trapezoid height of 2.83 feet, and a prism height of 5 feet.

- Determine which formula to use: (
*b*1+*b*2)*h*+*PH* - Plug in what information is given: (3+5)(2.83) + P(5)
- Figure out the rest: Need to find
*P*.*P*=3+3+3+5=14 - Plug in new information: (3+5)(2.83)+(14)(5)
- Perform mathematical operations: (8)(2.83) + 70 = 22.64+70 =92.64
- Give correct units: 92.64 square feet

Now let's take a look at our second example:

Find the volume of a trapezoidal prism that has bases of 4 and 5 feet, sides of 4 and 4.1 feet, a trapezoid height of 4 feet, and a prism height of 2 feet.

- Determine which formula to use:
*Hh*(*b*1+*b*2)/2 - Plug in what information is given: 2(4)(4+5)/2
- Perform mathematical operations: 8(9)/2
- Perform mathematical operations: 72/2 = 36
- Give correct units: 36 cubic feet

Let's review. **Trapezoids** are quadrilaterals (four sided figures) that have only one set of parallel lines, while **trapezoidal prisms** are three-dimensional solids that have a trapezoid for their top and bottom (their bases). They have four rectangles that make up the lateral faces. The formula for finding the surface area of the trapezoidal prism is Surface Area (*S.A.*)=(*b*1+*b*2)*h* +*PH*. The formula for finding the volume of a trapezoidal prism is *V*=*Hh*(*b*1+*b*2)/2, similar to the area of a trapezoid, which is simply *h*(*b*1+*b*2)/2.

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Basic Geometry: Help & Review16 chapters | 109 lessons

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