How Do You Find the Area of a Pentagonal Prism?

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Calculating the surface area of a pentagonal prism, especially one with irregular pentagons, can be a tricky, multi-step process. It requires a method of measuring edge lengths and angles and a calculator to add together the areas of the individual faces.

  1. Draw a diagram

    On a piece of paper, sketch the prism as if it were unfolded and laid flat. Your diagram should consist of five adjacent rectangles, all of which have equal lengths but possibly unequal widths. On each side of the stack of rectangles, there should be a pentagon. The pentagons should be mirror images, and each should have one edge connected to a rectangle of similar width.

  2. Measure the edges

    Using a ruler or some other measuring tool, find the measurements of each edge. Keep in mind that each edge of a pentagonal face is repeated in the pentagon on the other side. Also, the lengths of all five rectangles are equal.

  3. Calculate the surface area

    The surface area of any prism is a sum of the areas of the two parallel faces and the rectangular faces that connect them. For reach rectangle, the area is length times width. Because they all have the same length, you can add all of the widths together and multiply this sum by the length once. For a regular pentagon, the surface area is equal to 5/2 * s * a, where s is a side length and a is the apothem, or the height of the internal triangles. To find the area of an irregular pentagon, break it up into trapezoids and triangles. (You never need more than two trapezoids and one triangle.) Multiply the area of the pentagon by two to account for both faces. Add the rectangular and pentagonal faces' areas together.