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## Geometry for Beginners – How To Find the Surface Area and Volume of Prisms

Welcome to Geometry for Beginners. In this article, we are going to start looking at the 3-dimensional equivalents of the 2-dimensional relationships of perimeter and area. Perimeter of polygons refers to “the distance around” while the area of polygons refers to “the space inside.” For 3-dimensional figures, like prisms, “the distance around” becomes surface area; and “the space inside” becomes volume. We will be introducing the formulas for finding both surface area and volume of prisms and discussing applications of these concepts.

Before we discuss the formulas, we need to make sure we are all picturing the same kind of figure. Let’s use a cereal box for our mental image. Our cereal box is an example of a **right rectangular prism**. *RIGHT* because the sides (*lateral faces*) are perpendicular to the top and bottom (*bases*). *RECTANGULAR* because the *bases* (top and bottom) are *rectangles*. PRISM because the figure has two identical polygonal bases that are parallel to each other and 4 lateral faces that are rectangles.

Note: If the sides were not perpendicular to the bases, the figure would be OBLIQUE–not right, and the lateral faces would be parallelograms rather than rectangles. An oblique triangular prism would have sides NOT perpendicular to the bases, the bases would be triangles, and the 3 lateral faces would be parallelograms.

Again, picturing our cereal box, surface area would refer to the packaging or the box itself. For package manufacturers, the amount of material needed for each box is extremely important. The cereal inside the box would represent the volume of the package, assuming the box is full. This is an equally important business concern.

The formulas for surface area and volume will look unusual because they use symbols we haven’t used before, but only the symbols are new. You already have the skills to use these formulas!

**Formula for the Surface Area of Prisms: SA = 2B + LA**, where *SA is surface area*, *B is AREA of the base*, and

*LA is lateral area*.

Thus, to calculate the surface area of a prism, we must first calculate B, the area of whatever polygon forms the base, using the appropriate formula for the shape. Then, this value must be multiplied by two since there are 2 bases.

Next, we must calculate the lateral area which is the sum of the areas of the sides. Since the sides are either rectangles or parallelograms, we will use the formula A = bh. Then add the sides together for the lateral area. Caution! Be certain you are using the actual height and not a side if the sides are parallelograms.

The final step is to add your values of 2B and LA.

The best way to memorize and read the formula SA = 2B + LA is **“The surface area of a prism is equal to two times the area of the base plus the sum of the areas of all the lateral sides.”** Remember that area is always labeled as square units.

**Formula for the Volume of Prisms: V = Bh**, where, again, *B is the* * AREA of the base* and

*h is the height*of the prism.

Caution! Remember that the edge of the prism is the height **only if** the figure is a right prism. If the prism is oblique, the height will have to be calculated just as is necessary in non-right triangles.

The formula V = Bh should be read as** “The volume of a prism is equal to the area of the base times the height of the prism.”** Remember that volume is labeled as cubic units.

I think you can tell that these formulas are actually rather simple to calculate **IF** you have mastered the terminology and the area formulas for 2-dimensional polygons. The secret to success in Geometry: **Memorize! Memorize! Memorize!**

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