LENGTH, AREA, and VOLUME (Part I)
Length: When people think about measuring something, one image that quickly comes to
mind is the use of a ruler or tape measure to measure the length, width or height of
something. We say that the quantities of length, width, and height are all measured in
“units of length.” Another way of saying this is that they all have “dimensions of length.”

The inch is one “unit of length.” Give a few examples of some others:
Area is another quantity of great importance in all of science. By now you should have
measured the area of a piece of paper or the floor of a room (or both).

Imagine that somebody gave you a ruler and asked you to measure the area of one
side of this piece of paper in square inches. What would you do?

Imagine another student found the area of this piece of paper by placing one-inch
by one-inch tiles all over the paper, breaking the tiles into fractions when needed,
and then counting all of the tiles used. Would the student using this method get
(approximately) the same answer that you would get with a ruler? Explain in your
own words (and pictures?) why these or why not.

Could you measure area if you didn't know how to measure length? Discuss this
with your classmates based on the two measurement methods above (the “ruler
method” and the “tile method”). There is no right or wrong answer here. Record
your ideas below.
Volume is a measure of how much space something occupies.
 Imagine you had a rectangular block of wood and a ruler in front of you. How
would you find the volume of the block of wood?
 Remember the student who used square tiles to measure area? What would be the
equivalent of that “square tile method” applied to measurement of volume? How
could you make that work?
(By the way, volume and mass are not the same thing. If an iron bar and a wooden board
had exactly the same size, thickness, and shape then they would have the same volume.
The bar would have more mass because iron is much more dense than wood. We'll learn
more about mass later.)
When we measure length, area, and volume we are measuring different things. We have
already determined that the inch is a unit of length.

List a few units of area.

Are any units of area directly derived from the unit of length called the "inch"?
Which one and how is it related to the unit of length?

List a few units of volume.

Are any units of volume directly derived from the unit of length called the "inch"?
Which one and how is it related to the unit of length?
Check your answers with an instructor.
LENGTH, AREA, and VOLUME (Part II)
We are now going to investigate the relationships between length, area, and volume for
some three dimensional shapes. The arithmetic gets a little tricky and there are too many
shapes for every student to do all of them, so we will split into groups.
The groups will generate a lot of numbers, but the numbers themselves aren't important.
The groups will calculate areas and volumes using mathematical formulas, but you do not
need to memorize any formula. The point of this exercise is to look for patterns in the
areas, volumes, and numbers that we calculate. In order to do that, we will compare the
numbers and make graphs.
Students will split into three groups (counting off by threes works just fine). Each of the
three groups will look at a three-dimensional shape, but you can call yourself by the name
of the two-dimensional shape if you like.
The three groups will be
1. The sphere group (or circle group),
2. The cube group (or square group),
3. The tetrahedron group (or triangle group).
(You probably know what spheres and cubes are. Tetrahedrons are what you get if
you tape the edges of four triangles together, like pyramids with triangular bases.)
Each group will divide into three subgroups:
a) The perimeter group,
b) The area group,
c) The volume group.
Each subgroup will divide into two teams with two or three people in each.
Each subgroup will graph how their property (perimeter, area, or volume) of their shape
(sphere, cube, or tetrahedron) varies with a measure of length for their shape (the
diameter of a sphere or the length of a side of a square or a triangle). Within each
subgroup, the two teams should check their answers with each other to see that they are
doing the arithmetic correctly.
Each student should get a handout for preparing a table and a graph.
DATA TABLE: Circle or Sphere Group
Geometry books often describe the size of a sphere by the radius, r. The radius only
stretches halfway across the sphere. Since we are going to describe the other shapes by
the length of a side (which stretches all the way across), for this exercise we will describe
the size of a sphere by the diameter, d.
The first thing you need to know is how to find the circumference, the surface area, and
the volume of a sphere from the diameter. You can use the following mathematical
formulas. (Don’t worry about memorizing them! You don’t need to know these. We are
learning about length, area, and volume, not about spheres.)
Diameter = d
Don’t worry about where these formulas came from,
but trust us. They are right. They may look funny
since you may be used to seeing formulas for circles
using the radius, but these are correct.
Circumference = (3.14) d
Surface Area = (3.14) d2
Volume = (0.523) d3
Use these formulas to fill in the table below (you only need to work on the column that
you are responsible for, and you can copy the other results from your classmates.)
Diameter in cm
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Circumf. in ____
Surf. area in _____
Volume in ____
DATA TABLE: Cube or Square Group
Geometry books usually describe the size of a cube by the length of a side, s. That
description will work fine for us as well.
The first thing you need to know is how to find the perimeter of one face, the surface
area, and the volume of a cube from the length of a side. You can use the following
mathematical formulas. (Don’t worry about memorizing them! You don’t need to know
these. We are learning about length, area, and volume, not about cubes.)
Length of a side = s
Perimeter of one face = (4) s
Total Surface Area = (6) s2
Don’t worry about where these formulas came
from, but trust us. They are right. They may look
funny since you may be used to seeing formulas
for squares and not cubes, but these are correct.
Volume = s3
Use these formulas to fill in the table below (you only need to work on the column that
you are responsible for, and you can copy the other results from your classmates.)
Length of one side
in cm
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Perimeter of one
face in ____
Surface area (total)
in _____
Volume in ____
DATA TABLE: Tetrahedron or Triangle Group
Geometry books usually describe the size of a tetrahedron by the length of a side, s. That
description will work fine for us as well, but you may have trouble finding the formula
for the area or volume in terms of s. We will assume our tetrahedron is made entirely of
equilateral triangles. The correct formulas are given below.
The first thing you need to know is how to find the perimeter of one face, the surface
area, and the volume of a tetrahedron from the length of a side. You can use the
following mathematical formulas. (Don’t worry about memorizing them! You don’t need
to know these. We are learning about length, area, and volume, not about tetrahedra.)
Length of a side = d
Perimeter of one face = (3) d
Total Surface Area = (1.73) d2
Don’t worry about where these formulas came from,
but trust us. They are right. They may look funny
since you are probably used to seeing formulas for
triangles and not tetrahedra, but these are correct.
Volume = (0.816) d3
Use these formulas to fill in the table below (you only need to work on the column that
you are responsible for, and you can copy the other results from your classmates.)
Length of one side
in cm
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Perimeter of one
face in ____
Surface area (total)
in _____
Volume in ____
GRAPHING: Now make a graph of your property (perimeter, area, or volume) of your
shape (sphere, cube, or tetrahedron) as a function of your independent variable (the
length of a side or the diameter). Label the axes and give the graph a title. You may work
in groups on this, but each student should prepare a graph.
After you have made your graph, get together with the other people working on your
same shape and try to answer these questions…
Questions for Review:
Look at the properties of your shape when the length of a side (or diameter) was 1 cm and
when it was 2 cm. The length increased by a factor of 2.

The perimeter increased by a factor of ________________.

The surface area increased by a factor of ________________.

The volume increased by a factor of ___________________.
Now look at
the properties of your shape when the length of a side (or diameter) was 3 cm
and when it was 6 cm. Again, the length increased by a factor of 2.

The perimeter increased by a factor of ________________.

The surface area increased by a factor of ________________.

The volume increased by a factor of ___________________.
Now look at
the properties of your shape when the length of a side (or diameter) was 2 cm
and when it was 6 cm. This time the length increased by a factor of 3.

The perimeter increased by a factor of ________________.

The surface area increased by a factor of ________________.

The volume increased by a factor of ___________________.
Now look at
the properties of your shape when the length of a side (or diameter) was 2 cm
and when it was 8 cm. This time the length increased by a factor of 4.

The perimeter increased by a factor of ________________.

The surface area increased by a factor of ________________.

The volume increased by a factor of ___________________.
Answer the following questions in words (and symbols if you want).
If we measure the length of a side (or diameter) in units of centimeters, then …

We measure the perimeter (or circumference) in units of ________________.

We measure the surface area in units of ______________________________.

We measure the volume in units of __________________________________.
Get together with your classmates and see how these properties compare for different
shapes. For each of these shapes we assume that the larger object is the same shape
as the smaller object (so when a circle grows by a factor of two it is still a circle).
Perimeter:

Circle: if the diameter increases by a factor of two, the circumference
increases _________________________.

Cube: if the length of a side increases by a factor of two, the perimeter
increases _________________________.

Tetrahedron: if the length of a side increases by a factor of two, the perimeter
increases _________________________.
Area:

Circle: if the diameter increases by a factor of three, the surface area increases
_________________________.

Cube: if the length of a side increases by a factor of three, the surface area
increases _________________________.

Tetrahedron: if the length of a side increases by a factor of three, the surface
area increases _________________________.
Volume:

Circle: if the diameter increases by a factor of two, the volume increases
_________________________.

Cube: if the length of a side increases by a factor of two, the volume increases
_________________________.

Tetrahedron: if the length of a side increases by a factor of two, the volume
increases _________________________.
Now take a few minutes two write out in your own words, what is the pattern here?
Think of a baby elephant. Baby elephants are, um, elephant-shaped.
 What is the shape of an adult elephant?
As the baby elephant grows, the height of the elephant will eventually triple. When
the height triples…

What will happen to the length?

What will happen to the surface area?

What will happen to the volume?