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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:

rectangular prism

If we consider the front face (Rectangle ABCD) to be the base we can see that it has an area of 12 square units. There are four rows of this face, each with 12 cubes in it.

rect prism with lables

So, if we multiply the area of the face (12) by the 4 rows, we find that there are 48 cubes, or a volume of 48 cubic units!

Let's look at another type of prism: a cube! Here's a sample of the surface area and volume of a cube.

Now let's look at a cylinder:

cylinder

If the area of the circular base is equal to 16π square units, and there are five rows of these circular bases, then the volume would be 16π × 5 = 80π cubic units, or approximately 251.2 cubic units.

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

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 height
Volume = 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 2D shape. Large "B" is the area of the base of a 3D solid.

Example 1

rectangular prism #1

With a rectangular prism it doesn't matter which face you say is the base, your answer will turn out the same.

Let's say the bottom face is the base (the one it's sitting on):

Area of Base (B) = Bh = 9.0 x 5.4 cm

B = 48.6 cm^2


Example 2

Triangular Prism (#3)

With a triangular prism, the bases are the parallel sides (where the triangles are).

B = (1/2)bh = (1/2)(9.0 x 4.0 cm)

B = 18 cm^2


Example 3

prism with trapezoid base

The base of this prism is a trapezoid.

B = (1/2)(b1 + b2)h = (1/2)(4+10) x 2 cm

B = 14 cm^2


Example 4

Cylinder (#4)

A cylinder is pretty much a prism since it has parallel bases. In this example the diameter of the circle is given. As we discussed earlier, the radius is half the diameter.

B = (pi) x r^2 = (pi) x 2^2

B = 4(pi) cm^2 or approx = 12.56 cm^2


Exercise 1

Find the volume of a cube with sides of 7 cm.

Exercise 2

The area of the base of a hexagonal prism is 70.25 in2. It has a height of 10 inches. What is the volume?

Exercise 3

How much space is inside the cardboard center of a paper towel roll 12 inches long and with a diameter of 1 in?

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