Area Unit
Unit 3: Pre-Unit - Area
Geometry
Overview:
The purpose of this unit is to re-familiarize and introduce the concept of area to
students. While they should have already dealt with this, it is very possible that, with
new standards just being implemented, they could have not seen this information yet.
Formulas and their implementation will be the focus, but great attention will be given to
their application to real world problems. The unit will end with a general look at surface
area.
Student Goals:
• Find the area of common geometric figures using their formulas.
• Be able to derive area formulas using the formula for the area of a triangle.
• Honors: Be able to find areas of triangles using law of sines and law of cosines.
• Be able to draw the net of basic solid structures.
• Be able to find the surface area of basic solid structures.
• Be able to find areas in general application problems.
1
Area Unit
Day 1: Area Defined
Students will be able to define what area is in mathematical and non-mathematical frames of
reference.
Guiding Questions:
• What is meant by the term area?
• What unit can we use to measure area?
We use the term area frequently in normal day to day conversations, but we rarely stop to
think about what this term really means. What is area? How do we calculate it? Are there
systematic ways to determine the areas of shapes? This is what we want to explore today as we
begin our investigation into area.
To begin we will look at exploring area in a slightly unconventional way: with skittles. You and
a partner will get an activity packet (Activity P3-1A) and a bag of Skittles. You will explore the
connection between what area is as well as how we can get formulas to find the areas of basic
geometric shapes.
So, the aim of this activity was to arrive at a definition for area. Informally, we can define area
as the space within a given shape. Formally, area is the amount of two dimensional space
contained by an enclosed one dimensional structure.
2
Area Unit
Day 2: Formulas for Rectangles and Parallelograms.
Students will be able to determine the area of a rectangle and a parallelogram.
Guiding Questions:
• How many dimensions does area exist in?
• How are dimensions and exponents related to each other?
• Why are the area formulas for rectangles and parallelograms the same?
Motivating this unit will be how students can determine the area of an irregular figure such as
the state of Kentucky, a guitar front, or a piece of clothing. To this end they will need to become
familiar with the ways in which we can quickly compute the areas of basic geometric shapes.
So, we want to find the area of a shape, but, how do we measure area? We learned in the last
lesson that area measures the amount of space within a shape, but what is the unit that we
measure this space with? What is its basic form?
Our clue is in the definition itself: Area is the amount of two dimensional space bounded by
one dimensional space. When we measure one dimensional space we are measuring length. So,
we have inches, feet, meters, etc. Two dimensional space is the extension of one dimensional
space parallel to itself, giving us width. Thus, area - or two dimensional space - is a measure of
length and width, or more precisely length by width which is really length times width. This
means that we are measuring feet times feet, or feet squared; or we could be measuring meters
times meters or meters squared. The important aspect of any area unit is the squared part, the
exponent of 2. This refers to the dimensions we are representing, the two dimensions of area.
To measure area, any area, we begin by choosing which one dimensional unit we want: in this
example we will use centimeters. When we measure area using a unit of measure, we take one
unit of that measurement - one centimeter - and square it to produce a one centimeter by one
centimeter unit of area. This is the foundation to understanding area, that we are attempting to
calculate how many of these units of area fit into a shape. So, in essence, we are counting up
how many of these squares of unit area fit into the shape we are measuring.
To begin with, we start with the most basic shape to understanding area: the rectangle.
It will be necessary to reintroduce the definition of a rectangle to the class. Remind them that a
rectangle has four congruent 90˚ angles and opposite sides that are congruent and parallel.
Students will also be looking at parallelograms. Remind them that parallelograms are defined
as quadrilaterals with opposite sides that are parallel and congruent.
Because Geometry is more inclined to inquiry based learning, students will begin by exploring
the areas of various rectangles and parallelograms without formulas (Activity P3-2A). Instead,
they will be attempting to discover the areas for themselves and as a class attempt to arrive at
a formula for finding their areas. They also will look for the connections between the areas of
rectangles and parallelograms.
Important questions for them to consider as they investigate are:
• What methods can you use to find the area of rectangles and parallelograms?
• What is the simplest method that you can find to figure out these areas?
• Are there any similarities between rectangles and parallelograms and their areas?
Important questions for them to consider after they have investigated are:
• What relationship exists between parallelograms and rectangles that allows you to use the
same formula to find their areas?
• How can you make a rectangle into a parallelogram and vice versa?
• Why can you not use the side lengths to find the area of a parallelogram as you do with a
rectangle?
• How can area remain the same even when the shape changes?
3
Area Unit
During the post-activity discussion, make sure to call on students directly to share their
solutions to finding these areas. Ensure that it does not devolve into a didactic, one-at-a-time
environment. Encourage students to chime in if they tried the same methods or did something
similar. At the very least, make certain you are connecting the efforts of the students to one
another so that they can see there is no one right way to get the answer, and that there various
methods are all connected.
Finish by asking the students the aforementioned questions for after the discussion. Explain
how parallelograms can be rearranged into a rectangle and how a rectangle can also be
rearranged into a parallelogram. For the more intuitive students, introduce the idea of slicing
the rectangle into an infinite number of horizontal pieces and then pushing them to the side to
create a parallelogram. This can be a useful tool to explaining the tie between these shapes. As
a challenge, ask the students to explain how area can remain the same even when we change
the shape of a rectangle to one of its related parallelogram forms.
To end the lesson, formalize the definition for the formula of rectangles and parallelograms.
Formula for the areas of rectangles and parallelograms
To find the area of a rectangle or a parallelogram, you multiply the base times the height.
ARectangle = b × h
AParallelogram = b × h
where A is the area, b is the base, and h is the height.
Students will finish by applying the formula to their homework (Homework P3-2B).
Exit Slip: How many rectangles have the area of
50 m 2 ?
4
Area Unit
Day 3: Formula for Areas of Triangles and Trapezoids
Students will be able to find the area of a triangle and a trapezoid.
Guiding Questions:
• How are the areas of triangles and trapezoids related to the areas of rectangles and
parallelograms?
• How do we find the base of a triangle and a trapezoid?
The most important aspect to the study of area is the realization that all formulas for basic
shapes are essentially the same: base times height. This lesson will look to explore this as we
attempt to derive the formulas for a triangle and a trapezoid.
When finding the areas of triangles and trapezoids the central focus should be on the
similarities that these two shapes have. While not apparent at first glance, closer study reveals
their symbiotic nature. It may be apparent that a trapezoid has two bases to a student, but to
make the same claim for a triangle can be problematic. It is difficult for a beginning geometer
to think of a point, or vertex of the triangle, as a base in the normal sense. However, this is the
underlying equality that ties triangles and trapezoids together.
To begin with, students will explore how they can derive the formulas for a triangle and
trapezoidal area using their knowledge of rectangles and parallelograms. They will construct
these new shapes within the older shapes and then work towards a formula for their area.
The only introductions needed are definitions of the shapes in question.
A triangle is a polygon with three sides.
A trapezoid is a quadrilateral that has exactly one pair of opposite sides which are parallel.
Students will now begin their investigation. They will be given 3 triangles and 3 trapezoids
which they will have to find the area of. They will then be given two generic triangles with the
base and height labeled, and two generic trapezoids with the bases and height labeled, with
which they will be asked to prove the formulas for each are true. (Activity P3-3A)
After students have had time to explore, explain to them the connection between the area
formulas for trapezoids and triangles, as well as both of these area formulas to those of
rectangles and parallelograms. Particularly you should point out the following:
• The second base of a triangle has a measure of zero.
• The area formulas for triangles and trapezoids is essentially the average of the bases times
the height - suspiciously familiar to the formula for rectangles and parallelograms.
Finish by showing algebraic proofs of these formulas to honors classes, and intuitive
transformation proofs to regular classes.
Finish by passing out homework (Homework P3-3B) and writing exit slip on board or showing
on projector.
Exit Slip: What is the basic formula to remember when finding the areas of basic geometric
shapes?
5
Area Unit
Day 4: Areas of Circles
Students will be able to find the area of a circle.
Guiding Questions:
• What ratio does pi represent?
• How can we use triangles and parallelograms to derive the formula for the area of a circle?
So far students have looked at polygonal shapes. It is now time to connect these to the circle.
Students will do this by cutting a circle into pie-like pieces, and then use these pieces and
previous area formulas they know to approximate the area of a circle (Activity P3-4A).
Before they begin, time should be spent redefining a circle, specifically noting the relationship
between its diameter and radius as well as the meaning of circumference (perimeter of a circle)
and the formula for circumference. They also should be challenged to find why pi is everpresent when dealing with circles. Tell them it is a ratio, a constant, and they should attempt
to uncover its meaning.
diameter = 2 ⋅ radius
1
radius = ⋅ diameter
2
Circumference = 2π r
After they have had time to explore, show them how they can use their ideas to derive the area
formula for a circle.
Area Circle = π r 2
With the area formula in hand, do a few examples showing how to get the area of a circle with
a given radius. Also, a few examples should be done where the students must find the radius of
a circle with a given area or circumference. After this, they can begin to work on the homework
(Homework P3-4B).
Exit Slip: What ratio does pi represent?
6
Area Unit
Day 5: Areas of Irregular Shapes
Students will be able to find the area of irregular shapes.
Guiding Questions:
• How can I use my prior knowledge of areas to estimate the area of an irregular shape?
Now that students have the tools needed to find the areas of basic shapes, it is now time to
apply this knowledge to finding areas of irregular shapes.
If students are particularly adept at taking on new ideas they should be required to do the
following activities without examples. If they are weary, or not confident in their abilities, then
the following example should be shown.
Example: Find the area of the following shape.
4
To find the area of this figure, we must recognize the basic shapes of which is it formed.
There is a rectangle at the base, a triangle on the top right of this, and a half circle cut
out of the left. Therefore, to find the area of the figure, we must find the area of these
constituent smaller figures and combine them correctly.
1
⋅b ⋅h
2
1
= ⋅ (12 − 4 − 4 ) ⋅ ( 8 − 3)
2
1
= ⋅ ( 4 )( 5 )
2
= 10
Atriangle =
Arectangle = b ⋅ h
= 12 ⋅ 3
= 36
1 2
πr
2
1
2
= π (2)
2
= 6.3
Ahalf circle =
Ashape = Arectangle + Atriangle − Ahalf circle
= 36 + 10 − 6.3
= 39.7
7
Area Unit
Students will be given a handout that organizes the area formulas into an easy to read
organizer (Handout P3-5A). They will then use this handout to complete an activity based
around finding the area of irregular shapes. The activity culminates in finding the population
density of Kentucky and Newport. They then will have to explain the differences in the
numbers they find (Activity P3-5B).
Exit Slip: Find the area of an irregular shape and bring in your findings tomorrow.
8
Area Unit
Day 6: Nets and Surface Area
Students will be able to find the surface area of regular 3-dimensional shapes.
Guiding questions
• How is it possible to find a two dimensional measurement of a three dimensional object?
• What is a good way to describe what surface area measures?
To begin, students will have an initial homework or challenge to draw the nets of basic 3
dimensional shapes (Challenge P3-6A).
When going over correct responses, look for difficulties that students may have in visualizing
this process. Problems in this area tend to be magnified as they attempt to understand and
implement formulas for surface area and volume. Ask for students that were able to
accomplish this task to explain their reasoning and decision making processes in drawing the
nets.
Next, students should feel comfortable talking about the various types of basic 3-dimensional
shapes. To do this, hand out the graphic organizer on 3-dimensional shapes (Handout P3-6B)
and fill in this chart with them. Remind them that this sheet will be extremely helpful in the
successful completion of this objective.
Most important to tell the students when completing the handout is that naming solids starts
with the shape of the base and then ends with the way this base grows into 3-dimensional
space.
To introduce surface area, students will do an activity that requires them to explore finding the
surface area of the regular solid figures (Activity P3-6C). They will be required to generalize how
we find the surface area of these shapes, as well as what the specific formulas are for each
shape. This activity will have them use nets to determine the surface area.
Upon completion of the activity, have students write the surface area formulas of these shapes
in the handout they received earlier.
For homework they will be required to find a three dimensional object from home and draw the
net, name the solid, and find the surface area (Homework P3-6D). To go along with this, they
will have a short homework on naming and drawing solids, and finding their surface areas.
Upon entering the classroom the next day, choose a few students to share their findings from
home. Afterwards, go over the homework on naming and drawing solids, and finding their
surface areas. Ask students to write down their comfort level in naming and drawing solids on
a note card and then pass it in. Use this as a quick gauge of student understanding.
Students will now be asked to investigate the surface area of various platonic solids (Activity
P3-6E). To do this, they will be given one of three nets of platonic solids and then be asked to
determine the surface area of this shape. During their investigation they will be asked to write
about the steps they take to complete this task. Afterwards they will build their solid and then
hang it from the ceiling.
9
Area Unit
Day 7: Interior and Exterior Angles of a Polygon
Students will be able to find the measures of interior and exterior angles of polygons.
Guiding Questions
• Why do the exterior angles of any polygon always add up to 360˚?
• What is a regular polygon?
• What is the sum of the interior angles of a circle?
To begin with, discuss the definition of polygon. List the properties of polygons on the board
and have students come up and draw examples.
After this, explain the two types of angles that polygons have: interior and exterior.
Then draw a polygon on the board showing these two, making certain to explain how one
correctly draws the exterior angles of a polygon. Next, give the students 10 minutes to draw
there own polygons and explore what the sum of the exterior angles is. Challenge the students
to reach a general conclusion about what they are noticing. Also, do not limit their ability to
measure these exterior angles - let them use any method they desire. Once the ten minutes is
up point out particular students thoughts and conclusions to start a class discussion. Finally,
end with the Exterior Angle Theorem:
Exterior Angle Theorem
The exterior angles of all polygons will always add up to 360˚.
Demonstrate why this is so by illustrating how if we take a polygon and its exterior angles,
reducing the side lengths until they reach zero, we will get a pinwheel type pattern - a circle.
Show this process with polygons that have various numbers of sides.
For interior angles, begin by noting that there is no constant value for interior angles. Instead,
there is a formula. Introduce the Interior Angle Theorem:
Interior Angle Theorem
For a polygon with n sides, the sum of the interior angles of this polygon S can be found
using the formula S = ( n − 2 ) ⋅180 .
By corollary, if we want to find the measure of one interior angle, A, of a regular polygon with
n sides, we use the formula
A=
( n − 2 ) ⋅180 = 180 − 360 .
n
n
10
Area Unit
At this point have the students build a chart to display interior angle measures for successive
polygons with the following headings:
Measure of one interior angle of
regular polygon (A)
Number
of sides
(n)
n−2
3
1
180˚
60˚
4
2
360˚
90˚
5
3
540˚
108˚
Sum of Interior Angles (S)
S = ( n − 2 ) ⋅180
Have the students complete this chart to at least
questions as to the patterns they see.
A=
( n − 2 ) ⋅180 = 180 − 360
n
n
n = 10 sides. As they are doing this, ask
Show students how they can use this chart when doing problems. Go over various types of
problems implementing these theorems and then hand out homework (Homework P3-7A).
Exit Slip: Explain the Interior and Exterior Angle Theorems.
For the next day, have students fill in missing parts of each theorem.
11
Area Unit
Day 8: Geometric Probability
Students will be able to use area to find geometric probabilities.
Guiding Questions
• What is probability?
• How can we use area to find probability?
• What is a real life example of Geometric Probability?
To begin with, students need to understand the mathematical definition of probability. Urge
students to explain what probability means. You may need to reword it, using phrases such as
the likelihood, or the odds of, or the chances of. After a few moments of exploring, bring them
to a mathematical definition: Probability is the ratio of the number of times or ways you can be
successful over the total number of ways you could do something. Using the example of
flipping a coin may be helpful. After they have a grasp on the main idea of what probability is,
introduce the mathematical definition:
Definition of Probability
The probability that an event E will occur is the ratio of the number of successful ways E can
happen (S) over the total number of outcomes that can happen (n) when trying to perform E.
P(E) =
S
n
To begin with, introduce students to an easy activity demonstrating probability. Have them get
into groups of three and distribute one large sheet of paper and a ruler. The students will draw
a line on the paper and then try to find the probability of the ruler landing on this line when
dropping it from five feet above the paper. Have the students keep a written record of what they
do and the conclusions that they make. After having 10 minutes exploring, bring the class
back as a group, and then debrief over what they found. Keep track of which groups used the
definition of probability to guide their experiment and point out their methods to the rest of the
class. In particular, make sure the class sees that they should have been attempting to
measure one of two things; 1) the total number of hits to misses, or 2) the area of the region
that could produce a hit versus the area of the region where the ruler could possibly land.
Use this discussion as a segue to the discussion of geometric probability. Notably, you should
make the connection between area and probability. Use this time to give a few examples of how
one could calculate the geometric probability of different events.
12
Area Unit
Example 1:
Example 2:
Students can now begin their homework (Homework P3-8A).
Exit slip: What is geometric probability and how do you calculate it?
For the next day, have students give an example of a situation that uses geometric probability.
13
Area Unit
Activity P3-1A: What is Area
Name: ____________________________________
Directions:
• In this activity you will follow the directions and answer all questions given to you.
• You can work with a partner or independently.
• If you work with a partner, one will do the activity (the explorer), and one will write down
your answers and findings on these sheets (the recorder).
I. Triangles
Fill in the triangle on page A-3 with the skittles given to you and then answer the following
questions.
1. How many skittles does it take to completely fill this triangle? ___________________
2. How many skittles are there along the base of the triangle? ______________________
3. How many skittles high is your triangle? ________________________________________
4. What is the area of this triangle in skittles? Why did you choose this number?
5. If you multiply the number of skittles it takes to fill the triangle (Question 1) by 2 and
then divide this number by number of skittles along the base, what do you get? Does
this result look similar to the number of skittles high your triangle is? Why do you
think this is?
6. If you had a triangle that had 3 more skittles along the base and was 1 more skittle
high, how many skittles do you think it would take to fill this triangle? Explain your
answer.
II. Rectangles
Fill in the rectangle on page A-3 with the skittles given to you and then answer the following
questions.
1. How many skittles does it take to completely fill this rectangle? ___________________
2. How many skittles are there along the base of the rectangle? ______________________
3. How many skittles high is your rectangle? ________________________________________
4. What is the area of this rectangle in skittles? Why did you choose this number?
5. If you divide the number of skittles it takes to fill the rectangle by the number of skittles
along the base of the rectangle, what do you get? Does this result look similar to the
number of skittles high your rectangle is? Why do you think this is?
6. If you had a rectangle that had 2 more skittles along the base and was 4 more skittles
high, how many skittles do you think it would take to fill this rectangle? Explain your
answer.
A-1
Area Unit
III. Parallelograms
Fill in the parallelogram on page A-4 with the skittles given to you and then answer the
following questions.
1. How many skittles does it take to completely fill this parallelogram? ___________________
2. How many skittles are there along the base of the parallelogram? ______________________
3. How many skittles high is your parallelogram? ________________________________________
4. What is the area of this parallelogram in skittles? Why did you choose this number?
5. If you divide the number of skittles it takes to fill the parallelogram by the number of
skittles high the parallelogram is, what do you get? Does this result look similar to the
number of skittles along the base of your parallelogram? Why do you think this is?
6. How is the parallelogram similar to the rectangle you just worked with?
7. If you had a parallelogram that had 3 more skittles along the base and was 2 more
skittles high, how many skittles do you think it would take to fill this rectangle? Explain
your answer.
IV. Trapezoids
Fill in the trapezoid on page A-4 with the skittles given to you and then answer the
following questions.
1. How many skittles does it take to completely fill this trapezoid? ___________________
2. How many skittles are there along the bottom base of the trapezoid? ______________
3. How many skittles are there along the top base of the trapezoid? __________________
4. How many skittles high is your trapezoid? ________________________________________
5. What is the area of this trapezoid in skittles? Why did you choose this number?
6. Find the average of the two bases by adding them together and then dividing this
number by 2. What do you get?
______________________________________________
7. If you divide the number of skittles it takes to fill the trapezoid by the number of skittles
high the trapezoid is, what do you get? Does this result look similar to the average of
the bases of your trapezoid? Why do you think this is?
8. How is the trapezoid different from the other shapes we have worked with?
9. If you had a trapezoid that was 2 more skittles high, how many skittles do you think it
would take to fill this trapezoid? Explain your answer.
V. Closing Questions
1. What did you notice about the angle the height always made with the bases?
2. How would you define area?
3. Could you have done this activity with something other than skittles? How would it
have changed the outcomes?
4. Why might you want to measure area with different units other than skittles?
A-2
Area Unit
Triangle____________________________
H
E
I
G
H
T
BASE
Rectangle____________________________
H
E
I
G
H
T
BASE
A-3
Area Unit
Parallelogram____________________________
H
E
I
G
H
T
BASE
Trapezoid______________________________
TOP
BASE
H
E
I
G
H
T
BOTTOM
BASE
A-4
Area Unit
Activity P3-2A: Areas of Rectangles and Parallelograms
Name: _____________________
In this activity you will be exploring the areas of rectangles and parallelograms. You are given 8
figures to look at. You will have to find the areas of each figure using whatever method you can
think of. Remember, there are no wrong answers except not attempting one. Write down what
you do and see if you can develop a formula or easiest way to get the areas of these shapes.
B-1
Area Unit
Reflection Questions
1. What methods did you use to find the areas of these rectangles and parallelograms?
2. Which method of finding the area of these figures did you find the easiest to use? Why did
you prefer this method?
3. Describe any similarities between rectangles and parallelograms that you discovered. Did
these similarities help you in finding the areas of these shapes?
B-2
Area Unit
Homework P3-2B: Areas of Rectangles and Parallelograms
Name: _____________________
Find the area of each shape.
1.
2.
3.
4.
5.
6.
7.
8.
9. If you double the sides of a rectangle, how much bigger is the area?
10. If you double the area of a square, how much bigger are the sides?
B-3
Area Unit
Activity P3-3A: Triangles and Trapezoids
Name: ____________________________________
Part I
Find the area of the following triangles and trapezoids using any method you would like. As you
work, see if you can determine a formula for each shape, and also see if their are any similarities
between the two.
1. What methods did you use to find the areas of each figure? Which one worked best?
2. Were there any similarities between the way you found the areas for the triangles and the
areas for the trapezoids?
3. How do the areas you found relate to the areas of rectangles and parallelograms?
C-1
Area Unit
Part II
In this section you will be given the area formulas for both triangles and trapezoids and will try to
prove these are true given the following drawing for each. Be creative! Anything goes!
Area of Triangle
ATriangle =
1
b⋅h
2
b = base
h = height
Area of a Trapezoid
ATrapezoid =
1
h ( b1 + b2 )
2
h = height
b1 = base 1
b2 = base 2
C-2
Area Unit
Homework P3-3B: Triangles and Trapezoids
Name: _____________________________
Find the area of the following shapes.
1.
2.
3.
4.
5.
6.
7.
8.
9. Sketch and label a trapezoid that has an area of 100 cm2.
10. Change one number in the diagram you drew for the last question so that the area is now
200 cm2.
C-3
Area Unit
Homework P3-3B: Triangles and Trapezoids
Name: ******ANSWERS******
1. 28.8 yd2
2. 22.5 mi2
3. 29.7 mi2
4. 3.8 km2
5. 58.4 yd2
6. 23.46 m2
7. 28 km2
8. 22.68 yd2
9. Many answers.
10. Example: Double the height.
C-4
Area Unit
Activity P3-4A: Area of a Circle
Name: ____________________________________
What will I need?
- Paper
- Pencil
- Calculator
- Ruler
- Scissors
Why am I doing this?
- To investigate the area of circle and the meaning of pi, and how it represents a ratio
between a circle and its diameter.
Part I – Discovering the Area of a Circle (10 MINUTES)
1. Using the piece of paper given to you, use whatever method you can to create a
circle on the sheet.
2. Fold this sheet in half 3-4 times in a coffee filter pattern, then unfold and cut along
the creases.
3. Use the pie pieces you have created to figure out the area of the circle. Try to
arrange them into a shape that we have already studied.
4. Using the formula for circumference (the perimeter of a circle) to derive the general
area of a circle. (How are circumference and radius related to the length and height
of your arranged figure?)
Part II – Discovering the Value of Pi
1. Using the formula for circumference area that you found in step 4 above, find a
value for pi based on your measurements.
2. How close is your value of pi to the true value of pi? What could you have done to
get an even closer value to that of pi?
Part III – Archimedes Formula for Pi
1. From what we discussed on the board:
a. What is the difference between inscribing versus circumscribing?
b. What does it mean to say that Archimedes found the value of pi by finding the
upper limit and lower limit, effectively squeezing the value?
c. What is the value of pi using Archimedes method for an 8 sided figure? What
about a 16 sided figure.
D-1
Area Unit
Homework P3-4B: Areas of Circles
Name: ____________________________________
D-2
Area Unit
D-3
Area Unit
Homework P3-4B: Areas of Circles
Name: ******ANSWERS******
D-4
Area Unit
D-5
Area Unit
Handout P3-5A: Area Formulas
Name: ____________________________________
Fill in the following chart with the area formulas for the given shapes.
Shape
How do you find the area?
Area Formula
Rectangle
Multiply the base times the
height.
A = b⋅h
Parallelogram
Triangle
Trapezoid
Circle
E-1
Area Unit
Activity P3-5B: Areas of Irregular Shapes
Name: _____________________________
Now that you are an expert at finding areas, it is time to put your skills to the test. Below you
will be asked to find the areas of multiple irregular shapes, and then you will apply this new
skill to answering a question about population density.
Part I: Find the Areas of Irregular Shapes
E-2
Area Unit
Part II: Flooring Problem
A new building is to be constructed in Cincinnati this year. Most important to the owners is
that it have a breathtaking entrance. The floor design for the entrance looks like this:
where the measurements are in meters. They want to have marble floors for this space, and
have priced marble floors at $57.20 a square meter. How much will it cost them to cover this
floor in the marble?
Part III: Population Density
Population density is a measure of the amount of people that live in a certain amount of land
area. Given that the population of Kentucky is 4,314,113 people, and the population of
Newport is 15,273 people; use the maps to determine and compare the population density of
each.
1. What is the population density of Kentucky?
2. What is the population density of Newport?
3. How do the two values compare? What accounts for this difference?
4. What kind of population density do you think determines an urban area? Defend your
answer using your results.
5. Considering your findings, is Kentucky a more urban or rural state? Explain your choice.
E-3
Area Unit
E-4
Area Unit
Activity P3-5B: Areas of Irregular Shapes
Name: ******ANSWERS******
Now that you are an expert at finding areas, it is time to put your skills to the test. Below you
will be asked to find the areas of multiple irregular shapes, and then you will apply this new
skill to answering a question about population density.
Part I: Find the Areas of Irregular Shapes
A = 23.4cm 2
A = 272cm 2
A = 63.8cm 2
A = 259.7cm 2
A = 1847.3cm 2
A = 157.1cm 2
E-5
Area Unit
Part II: Flooring Problem
A new building is to be constructed in Cincinnati this year. Most important to the owners is
that it have a breathtaking entrance. The floor design for the entrance looks like this:
where the measurements are in meters. They want to have marble floors for this space, and
have priced marble floors at $57.20 a square meter. How much will it cost them to cover this
floor in the marble?
$36440.96
Part III: Population Density
Population density is a measure of the amount of people that live in a certain amount of land
area. Given that the population of Kentucky is 4,314,113 people, and the population of
Newport is 15,273 people; use the maps to determine and compare the population density of
each.
1. What is the population density of Kentucky?
106.8 people per square mile.
2. What is the population density of Newport?
5091 people per square mile.
3. How do the two values compare? What accounts for this difference?
The Newport value is 50 times bigger. Newport is a strictly urban area.
4. What kind of population density do you think determines an urban area? Defend your
answer using your results.
Answers will vary.
5. Considering your findings, is Kentucky a more urban or rural state? Explain your choice.
Answers will vary, but should conclude Kentucky is a more rural state.
E-6
Area Unit
Challenge P3-6A: Determine the Net of a Solid
Name: _____________________________
In this challenge you will be working with solids, or 3 dimensional objects.
Nets
Suppose that we wanted to use a “pattern” to create a solid in the same way that a
dressmaker uses a pattern to make a dress. Such a pattern is called a net.
A net is a two dimensional pattern for a solid.
Imagine being able to “unfold” each of the solids below, and draw a possible net for each
solid. Assume that all bases are regular polygons. (If you can build these shapes and then
unfold them, it may be very helpful.)
F-1
Area Unit
Handout P3-6B: Types of Solids and Their Surface Areas
Name: _____________________
Complete the table below concerning solids.
Solid Name
Picture
Surface Area Formula
Prism
Pyramid
Cylinder
Cone
Sphere
F-2
Area Unit
Activity P3-6C: Surface Areas of Solids
Name: _____________________________
For this activity you will be required to find the surface area of regular 3-dimensional objects.
Part 1: Prisms
1. How would you find the surface area of a rectangular prism?
2. Try to write a formula that will find the surface area of any rectangular prism.
3. How would you find the surface area of a triangular prism?
4. Try to write a formula that will find the surface area of any triangular prism.
5. What are the similarities and differences between the surface areas of rectangular prisms
and triangular prisms?
Part 2: Pyramids
1. How would you find the surface area of a rectangular pyramid?
2. Try to write a formula that will find the surface area of any rectangular pyramid.
3. How would you find the surface area of a triangular pyramid?
4. Try to write a formula that will find the surface area of any triangular pyramid.
5. What are the similarities and differences between the surface areas of rectangular pyramids
and triangular pyramids?
Part 3: Cylinders
1. How would you find the surface area of a cylinder?
2. Try to write a formula that will find the surface area of any cylinder.
3. What are the similarities and differences between the surface areas of cylinders and prisms?
Part 4: Cones
1. How would you find the surface area of a cone?
2. Try to write a formula that will find the surface area of any cone.
3. What are the similarities and differences between the surface areas of cones and pyramids?
Part 5: Spheres
1. How would you find the surface area of the sphere?
2. Try to write a formula that will find the surface area of any sphere.
3. What are the similarities and differences between the surface areas of spheres and all other
shapes?
F-3
Area Unit
Rectangular Prism
Triangular Prism
Rectangular Pyramid
Triangular Pyramid
F-4
Area Unit
Cylinder
Cone
Sphere
F-5
Area Unit
Homework P3-6D: Surface Areas of Solids
Name: _____________________________
For the following problems; 1) name the solid, 2) draw the net of the solid, and 3) find the
surface area.
1.
2.
3.
4.
5.
6.
F-6
Area Unit
7.
9.
11.
8.
10.
12.
13. A sphere with a diameter of 6.2 in.
14. A pyramid with slant height 6.8 mi whose triangular base measures 11 mi on each side.
Each altitude of the base measures 9.5 mi.
15. A prism 2 m tall. The base is a trapezoid whose parallel sides measure 7 m and 3 m. The
other sides are each 4 m. The altitude of the trapezoid measures 3.5 m.
F-7
Area Unit
Homework P3-6D: Surface Areas of Solids
Name: ******ANSWERS******
1. 144 square kilometers
2. 659.73 square kilometers
3. 238 square yards
4. 87.97 square centimeters
5. 143 square yards
6. 978 square yards
7. 187.7 square centimeters
8. 66.4 square inches
9. 113.1 square feet
10. 452.4 square centimeters
11. 885.9 square meters
12. 2029.5 square centimeters
13. 120.8 square inches
14. 164.5 square miles
15. 71 square meters
F-8
Area Unit
Activity P3-6E: Platonic Solids
Name: ____________________________________
For this activity, you will be given one of three platonic solids.
1. Explain how you could find the area of the solid you have.
2. Find the area of your solid and show your calculations below.
3. Describe what you think this shape will look like when you put it together.
4. Put your solid together and describe how the solid you made compares to what you thought
it would be.
F-9
Area Unit
F-10
Area Unit
F-11
Area Unit
F-12
Area Unit
Homework P3-7A: Interior and Exterior Angle Theorems
Name: _____________________
G-1
Area Unit
G-2
Area Unit
Homework P3-7A: Interior and Exterior Angle Theorems
Name: ******ANSWERS******
G-3
Area Unit
G-4
Area Unit
Homework P3-8A: Geometric Probability
Name: ____________________________________
1.
Find the probability that a point chosen at random falls within the shaded region.
2.
3.
4.
5.
A regular hexagonal shaped target with sides of length 14 centimeters has a circular bull’s eye
with a diameter of 3 centimeters. In problems 6-8, darts are thrown and hit the target at
random.
6. What is the probability that a dart that hits the
target will hit the bull’s eye?
7. Estimate how many times a dart will hit the bull’s
eye if 100 darts hit the target.
8. Find the probability that a dart will hit the bull’s
eye if the bull’s eye’s radius is doubled.
H-1
Area Unit
9. A circle with radius 2 units is circumscribed about a square with side length 2 units. Find
the probability that a randomly chosen point will be inside the circle but outside the
square.
In problems 10 and 11, use the following information.
A ship is known to have sunk off the coast, between an island and the mainland as shown. A
salvage vessel anchors at a random spot in this rectangular region for divers to search for
the ship.
10. Find the approximate area of the
rectangular region where the ship sank.
11. The divers search 500 feet in all
directions from a point on the ocean
floor directly below the salvage vessel.
Estimate the probability that the divers
will find the sunken ship on the first
try.
In problems 12-16, use the following information.
Imagine that an arrow hitting the target shown is equally likely to hit any point on the target.
The 10-point circle has a 4.8 inch diameter and each of the other rings is 2.4 inches wide.
Find the probability that the arrow hits the region described.
12. The 10-point region.
13. The yellow region.
14. The white region.
15. The 5-point region.
16. Does the geometric probability
model hold true when an expert
archer shoots an arrow? Explain
your reasoning.
H-2
Area Unit
Find the value of x so that the probability of the spinner landing on a blue sector is the value
given.
17.
1
3
18.
1
4
19.
1
6
In problems 20–22, use the following information.
In a “Hare and Hounds” balloon race, one balloon (the hare) leaves the ground first. About ten
minutes later, the other balloons (the hounds) leave. The hare then lands and marks a
square region as the target. The hounds each try to drop a marker in the target zone.
20. Suppose that a hound’s marker dropped onto a
rectangular field that is 200 feet by 250 feet is
equally likely to land anywhere in the field. The
target region is a 15 foot square that lies in the
field. What is the probability that the marker
lands in the target region?
21. If the area of the target region is doubled, how
does the probability change?
22. If each side of the target region is doubled, how
does the probability change?
23.
H-3