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Instruction Formulas |  Perimeter Formulas | Area Formulas | Three Dimensional Shapes and Surface Area | Volume Formulas | More Volumes and Areas | Formulas Involving Angles | Pythagorean Theorem

More Volumes and Areas
CA GR6 AF 2.1 & 3.2, CA GR6 MG 1.3, CA GR7 MG 2.1 & 2.2 & 2.3

Volumes and areas have special relationships that can be used to solve problems. Let's continue in understanding volumes and areas. 

Converting Units

Sometimes you will be asked to convert units to find the value of a problem. One inch is equal to 2.54 centimeters and one foot is equal to12 inches. When finding areas and volumes the conversions are different because we are using different dimensions. Observe the following example:

We know the area of the rectangle is (5 in) (3 in) = 15 sq in. However, what if you needed to know how many square feet the rectangle allows? Know that:

1 square foot = 144 square inches

So, 15 sq in. x 144 sq in. = 2,160 square inches.

Now let's look at the following 3D model.

See if you can find the volume in cubic centimeters (cm3)

Answer:

(6 in) x (5 in) x (3 in) = 90 in3

1 cubic inch = (2.54)3 cubic centimeters 16.39 cubic centimeters

So, 

90 in3. x 16.39  = 1,475.1 cm3.

Complex and Irregular Shapes

Not all shapes look simple. To find the area or volume of a shape, it can be necessary to break the shape down into simpler components. For example, below is a typical rectangle. However, if we rotate or view of the object, we can see that the object could easily be the front side of a prism - or even a cylinder!

Just the same, a problem could be presented in a manner you may not recognize right away. By combining shapes a new object can be formed. How would you find the volume of the following object?

We need to break this object down into two different shapes that we know well. In this case there is a triangular solid on top of a rectangular solid. All we need to do is find the volume of the top and add it to the volume of the bottom.

If we know the surface area of an object, we can find it's surface area if the object doubled. All you do is multiply the surface area by two squared or 22. See the example below:

We can do the same thing for volumes. Instead of squaring the number, we cube it.

The following example will show you:


 

Now, let's stop and take Practice Exercise 9-4. (top)

 

Next Page: Formulas Involving Angles