Study Guide

Volume of Prisms & Cylinders

Volume of Prisms & Cylinders

The volume of a solid is the amount of space inside the object. It's how much water fits inside a bathtub, how much sand fills a bucket, or how much soda your friend can chug and hold in his stomach.

Take a look at this rectangular prism. The volume is the number of unit blocks that fill the prism.

If we consider the front of this rectangular prism to be the base, we can see that it is made up of 12 cubes. There are four rows with 12 cubes in each row.

So, if we multiply the number of cubes in the base (12) by the number of rows (4), we find that there are 48 cubes. The prism has a volume of 48 cubic units.

We can also find the volume by measuring. The base of each cube is 1 unit wide and 1 unit high so its area is 1 unit2. Since there are 12 cubes in the base of the prism, the area of the prism base is 12 units2. The length of one cube is 1 unit, so the total length of the prism is 4 units. Notice that if we multiply the total area of the base (12 units2) by the total height of the base (4 units) we also get the area: 48 cubic units, or 48 units3.

Volume of a rectangular prism = (area of base) (height)

Let's look at another type of prism: a cube. The length, width, and height of a cube are all equal. It's a close relative to the square. A piece of origami paper (think thin, colorful, easy to fold) is a square, and a stack of origami paper is a cube if it's as high as the paper is wide (or long).

If each piece of origami paper is 5 inches long and 5 inches wide, the area of the piece of paper on the base of the stack is 25 inches2. In other words, the area of the base is 25 in2. If the stack is 5 inches high, then it has a volume of 25 in2 × 5 inches of paper, or 125 in3.

If we slide the top of the stack over, but the base stays where it is, we can make a leaning tower of Pisa, or an oblique prism. While the shape of the prism has changed a bit, the volume of the prism has not. The same amount of paper is still there. The volume of an oblique prism is also (area of base)(height), but the height has to be perpendicular to the base. In the case of the mysterious leaning tower of origami, the height is the distance from the top piece of paper straight down to the surface it sits on.

Volume of a square prism, or cube = (area of base)(height)

Not all prisms have a rectangular or square base. Think about those cool rainbow prisms you've seen in science.

This is a triangular prism. The bases are triangles and the sides (or faces) are rectangles. It's just like the stack of origami paper, only the paper is now triangle shaped. And the volume formula is the same as well, the area of the triangle base (either one) is multiplied by the triangular prism's height.

Volume of a triangular prism = (area of base)(height)

Now let's look at a cylinder.

If the area of the circular base is equal to 16π square units, and each row is 1 unit high, with five rows of these circular bases, then the volume would be 16π units2 × 5 units = 80π cubic units, or approximately 251.2units3.

Volume of a cylinder = (area of base)(height)

Look Out: volume is always measured in cubic units (units3). This is because we are dealing with three-dimensional objects now. You're in the big time!

See the pattern in these formulas? They all multiply the area of the base by how tall the 3-D shape is. This is pretty much all you have to do to find the volume of any prism or cylinder: find the area of the base and multiply it by the height.

Volume of a Prism or Cylinder = area of the Base x heightVolume = Bh

Look Out: note the difference between small "b" and large "B". In the examples above (and often in geometry in general), small "b" is the length of the base of a 2-D shape. Large "B" is the area of the base of a 3-D solid.