If you need to make real objects from your virtual 3-D designs, you’ll need to find the object’s surface area. And that’s because surface area is used to determine the quantity of materials needed. If you know the surface area of a box, you know how much cardboard is needed to build it. If you know the surface area of a roof, you can calculate the number of shingles needed to cover it. And if you know how to calculate the surface area of a 3-D trapezoid, also known as a trapezoidal prism, you know how to determine a wide variety of shapes used almost everywhere.

- Skill level:
- Moderate

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### Things you need

- Rectangular Prism Formula Sheet
- Analytical Geometry Book
- Paper and Pencil
- Calculator

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## Instructions

- 1
Determine the critical dimensions of each face of the 3-D trapezoid. The 3-D trapezoid has six faces. Two of the faces are rectangles, often called the lateral faces. The other four faces are trapezoids, specifically 2-D trapezoids.

For each of the two lateral faces the critical dimensions are the dimensions needed to calculate the area of each rectangle; these are the width and length. For a rectangle the formula for the area is the width multiplied by the length.

For each trapezoid face, the necessary dimensions include the lengths of the sides that are parallel, also known as the bases, and the distance between each parallel line on each face. The latter is the trapezoidal face height, also known as the altitude. The altitude is perpendicular to the trapezoid's baselines.

- 2
Calculate the total area of the lateral faces. First calculate the area of each of the lateral faces. To do this, multiply the width by the length of each lateral face. Next add these two areas. For example, if the width of the bottom lateral face was 2 inches and it had a length of 3 inches, the area would be 6 square inches. If the width of the top lateral face was 1 inch and the length was 2 inches, the area would be 2 square inches. The total area would then be 6 plus 2 or 8 square inches.

- 3
Calculate the total the area of the four trapezoidal faces. For each trapezoid face add the lengths of the bases. Then divide this sum by two. Then multiply the result by the altitude of the trapezoid. This result is the surface area of the face.

For example, if a trapezoid face has a base of 1 inch and another base of 2 inches and an altitude of 3 inches, the sum of the bases is 1 plus 2, or 3 inches. Three inches divided by 2 inches is 1.5 inches. Multiplying 1.5 inches by the altitude of 3 inches yields 4.5 inches, for an area of 4.5 square inches.

Now repeat the process for each of the remaining three trapezoid faces. After you obtain the area for each trapezoid face, add the four area calculations. This sum is the total area of the four trapezoidal faces.

- 4
Add the total area of the lateral and trapezoidal faces. Take the sum of the two lateral faces and the sum of the four trapezoid faces and add them together. This is the total surface area of the 3-D trapezoid.