ATM - Area and Volume Lesson

 

Area and Volume

This module in the SAT prep is about formulas and the interpretation of formulas to solve problems about objects.  A formula sheet Links to an external site. is available to you for this module here and on your SAT test.  Please print this page and use it with the lessons and quizzes in this module. These formulas do not need to be memorized, but you must know how to use them to find different dimensions of the various figures involved.  

As we start this lesson, let's make sure that we understand the difference between area and volume.  

Area is two-dimensional. The area is the space that a flat object takes up. The rectangle below has the area taken up by the blue space. The triangle area is the space taken up by the green triangle. The area of the arrow is the space taken up by the navy arrow.   Area dimensions mean the units are squared: feet², inches², kilometers^2, etc. For instance, the area of the rectangle is length times width giving the dimensions squared.

Note that the arrow is made up of a rectangle and a triangle, so when finding the area of the arrow, you could add the area of the rectangle plus the area of the triangle.

Volume is three dimensional. Volume is the space taken up within an object. The rectangle is converted to a rectangular box. The volume dimensions of the box are cubed, feet³ , inches³ , kilometers³  etc. To find the volume of the box, length times width time height is needed, three measures of the box, giving dimensions cubed.

Did you know that the rectangular box is also called a rectangular prism?  A prism is multi-sided object that consists of a consistent two dimensional object throughout its length. The rectangular box has a rectangle as its main two dimensional item of interest.

What about the cylinder in green above? How would we get the volume of this object? Examining the cylinder, we notice that it is obviously a circle on the top, so it is built of circles with height added as the third dimension. So the volume of a cylinder is the area of the circle (a two dimensional object) times the height. It is a type of prism, but since it is circular it has a special name, a cylinder.

What is similar between finding area and finding volume of objects?  Both are finding the amount of space that that an object takes up. The two dimensional object takes up flat 2D space, square units, and the three dimensional object takes up space within the object, 3D space, units cubed.

This section is manipulating formulas to solve dimensional problems. In other words, the volume may be given and a portion of the dimensions and you must find the missing dimension. We've worked with finding equivalent formulas earlier, and this is essentially the same. Solve the formula for a different dimension and input what is known to get the answer needed.

Example 1:   Find the height of a pyramid with a volume of 250 cubic meters and a square base length of 5 meters? 

Knowing that the shape is a pyramid the following item from the formula sheet should be selected for use from the SAT formula sheet Links to an external site..  Note that the formula is provided for you.

Now decide on the givens.  We are given the following:

                                                                      V = 250 meters³

                                                                      ℓ  = w  = 5 meters  

Note that the reference to a square base indicated that the sides of the base would be equal. The base in the picture is length and width so the dimension for length and width must be the same.

Understanding the parts that are given, substitute the values into the equation, and solve for the height h.

 $V=\frac{1}{3}lwh$

Method 1:   Use the equation as it is and substitute in the values.

$250=\frac{1}{3}\left(5\right)\left(5\right)h$

750 = 25h                               Multiply by 3 to remove the denominator.

30 = h                                         Divide by 25.

So the height of the pyramid with the square base is 30 meters.

Method 2:    Solve for h first.

h =         Multiply by 3 and divide by ℓ w.

h =   Substitute in the values that you have.

h =  

h =   30

Both methods show the answer of 30 meters for the height.

Now let's examine an object that is not on the formula sheet.  How would we approach this object using what we know that is on the formula sheet?

 

Example 2:  Find the volume of a triangular prism with the dimensions shown in the diagram.  The sides are left unshaded so that you can see that the object is built with many triangles from beginning to end of the span of 50 cm.

Note, pictures may or may not be provided. For this picture, I left the sides unshaded so that you can see that the object is built with many triangles from beginning to end of the span of 50 cm. The triangle is called the cross-section of the object if the object is cut at any place vertically through the object, a triangle will be seen on the cut edge.

To complete the problem, decide on the formula.  There is not one exactly like this on the formula sheet, but understanding that the basis for the object is a triangle, find the area formula for the triangle on the SAT formula sheet Links to an external site..  

The area formula here will help us create the formula for the triangular prism.  

Noting that the cross-section is a triangle to get the volume we add the dimension of the separate height of the prism. So the formula that we will use to solve the problem is

V = area of the triangle times the length of the prism.

We are given the base and height of the triangle in the diagram as well as the length, ℓ  , of the prism.  Note any letter may be used to represent the length, even x.  Each variable letter must be unique though if it stands for a different part.

$V=\frac{1}{2}bh\cdot l$

V = (1/2)(10)(35)(50)                      Substitute in the values.

V = 8,750 cm³                                            Note that the units are cubed. The base, height, and length were all in cm so cm*cm*cm = cm³

It is important to realize that with the basic building blocks you are able to build new formula equations on your own to answer questions about objects and ideas that you do not have a ready-made formula for.  Be open to looking for the clues and then putting the clues together to form the answer.  Here we realized the triangle was the key.

Now it is your turn to try some quick items to see if you can figure out the volume or the size of a part of the object.  Remember to use your formula sheet. To view the detailed solutions, download this handout Links to an external site..

 

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