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# 3.10: Perimeter and Area

Created by: CK-12

## Triangles and Parallelograms

I. Section Objectives

• Understand the basic concepts of the meaning of area.
• Use formulas to find the area of specific types of polygons.

II. Cross- curricular-Reflecting Pool dimensions

• Use the following image from Wikipedia of the Reflecting Pool in Washington, DC.
• This is Figure 10.01.01
• www.en.wikipedia.org/wiki/File:Reflecting_pool.jpg
• Here is the problem.
• According to Wikipedia, the dimensions of the Reflecting Pool are $2029\;\mathrm{ft}$ long and $167\;\mathrm{feet}$ wide.
• Given this information, what shape is the Reflecting Pool?
• What is the perimeter of the pool?
• What is the area of the pool?
• Solution:
• The shape is a rectangle.
• The perimeter is $2029 + 2029 + 167 + 167 = 4392\;\mathrm{ft.}$
• The area is $2029 \times 167 = 338,843\;\mathrm{sq. \ ft.}$

III. Technology Integration

• There are two great short videos on this website for area and length.
• One is an architect and one is on an apartment design.
• www.thefutureschannel.com/hands-on_math/apartment.php
• Have students watch the videos.
• Then you can expand on this by having the students draw a design of their room at home.
• Students will need to go home and do some measurements and then come back with the area and perimeter of their room.
• Rooms with unconventional shapes will be the most fun and challenge.
• Allow time for students to share their work.

IV. Notes on Assessment

• Assess student work on the Reflecting Pool problem.
• Is the diagram accurate?
• Did the students calculate the area correctly?
• Did the students calculate the perimeter correctly?
• Provide students with feedback on their work.

## Trapezoids, Rhombi and Kites

I. Section Objectives

• Understand the relationships between the areas of two categories of quadrilaterals: basic quadrilaterals and special quadrilaterals.
• Derive area formulas for trapezoids, rhombi and kites.
• Apply the area formula for these special quadrilaterals.

II. Cross- curricular-Room Design

• Tell students that they are going to design a room that has a trapezoidal shape.
• Students can complete this in connection with the Technology Integration if you choose.
• If not, have the students use the dimensions of their own bedroom (they did this in the last lesson), or the classroom or a standard size bedroom $(11 \times 10)$ for example.
• Students are going to redesign this area as a trapezoid.
• They want to come as close to the original area as possible.
• So if the room was $11 \times 10$, the area is $110\;\mathrm{sq\ feet.}$
• How can you come close to the same area if the shape of the room is a trapezoid?
• Students should draw their design on grid paper and explain their thinking.
• Allow time for students to share their work when finished.

III. Technology Integration

• This is a website that shows a house designed as a trapezoid.
• www.momoy.com/2009/04/02/l-house-beautiful-trapezoid-house-design-by-philippe-steubi-architekten-gmbh/
• Students can look at the trapezoid shape of the house and the floor plan is also included.
• There are views of the inside of the house and the outside of the house as well as some of the rooms.
• Conduct a discussion about the house. What would be the challenges of designing and building such a house?

IV. Notes on Assessment

• Assessment will come with student presentations and work product.
• What did students learn about the relationship between rectangles and trapezoids?
• Were they able to come up with a room with an area close to the original?
• Who got the closest?
• Provide students with feedback on their work.

## Area of Similar Polygons

I. Section Objectives

• Understand the relationship between the scale factor of similar polygons and their areas.
• Apply scale factors to solve problems about areas of similar polygons.
• Use scale models or scale drawings.

II. Cross- curricular-National Mall Mapping

• Ask students to use the Wikipedia image of the National Mall to create a map of it.
• This is Figure 10.03.01
• www.en.wikipedia.org/wiki/National_Mall
• Then tell the students that the mall is $1.9\;\mathrm{miles} \times 1.2\;\mathrm{miles.}$
• They are going to use what they have learned about scale and measurement to create their own map of the mall.
• They need to choose a scale to work with.
• Then they use grid paper to design the mall.
• When students have the area of the mall correct, they can draw in as many different museums and monuments as they can.
• Extra details add extra credit to their work.
• When finished, allow time for students to share their work.

III. Technology Integration

• Use the following website on the National Mall in Washington DC.
• www.en.wikipedia.org/wiki/National_Mall
• Have students complete some research about the mall.
• Possible questions include:
• Who designed it?
• When was it built?
• What is at the North end?
• What is at the South end?
• How many different museums can you visit there?
• Have you been to the mall?
• Which museum would you most like to visit or did you enjoy and why?

IV. Notes on Assessment

• Assess each student map.
• Is the use of scale done correctly?
• Are the measurements correct?
• Is the map accurate?
• Has the student take the time to add in details?
• Provide students with feedback on their work.

## Circumference and Arc Length

I. Section Objectives

• Understand the basic idea of a limit.
• Calculate the circumference of a circle.
• Calculate the length of an arc of a circle.

II. Cross- curricular-The Pantheon

• Have students use the image of the floor plan of the rotunda of the Pantheon to calculate the circumference of it.
• This is Figure 10.04.01
• www.en.wikipedia.org/wiki/Pantheon,_Rome
• The diameter of the dome is $142\;\mathrm{ft.}$
• Given this measurement, what is the circumference?
• Have the students draw a diagram to explain their work.
• Allow time for students to share their diagrams in small groups.

III. Technology Integration

• Have students use the following Wikipedia site to research information on the Pantheon.
• www.en.wikipedia.org/wiki/Pantheon,_Rome
• Students can use this information to write a short essay.
• Students should hunt for mathematical information about the Pantheon for their essay.
• For example, height of the columns.
• What is a portico?
• What is a rotunda?
• Have the students complete this work and then collect it for your review.
• Extension on initial exercise- have students research the dimensions of the rectangle that connect the portico and the rotunda.
• What is the area of the rectangle?
• What is the perimeter?

IV. Notes on Assessment

• Look at student work.
• Is it accurate?
• Does the diagram represent student work?
• Provide students with feedback on their work.

## Circles and Sectors

I. Section Objectives

• Calculate the area of a circle.
• Calculate the area of a sector.
• Expand understanding of the limit concept.

II. Cross- curricular-History

• Use the following image from the round table used by King Arthur.
• This is Figure 10.05.01
• The diameter of the round table was $18\;\mathrm{feet.}$
• Given this measurement, calculate the area of the round table.
• If the table was divided between each of the knights evenly, what is the area of one of the sectors?
• Draw a diagram to explain your work.
• Allow students time to share their diagrams when finished.

III. Technology Integration

• Have students use the following website as a tutorial on area and circumference of circles.
• www.mathgoodies.com/lessons/vol2/circle_area.html
• Students can review already learned material.
• There is also a worksheet section for them to work with and practice solving problems.

IV. Notes on Assessment

• Examine student diagrams.
• Were they able to find the correct area of the table?
• Does the diagram accurately show their work?
• Is there anything missing?
• Provide students with feedback/correction on their work.

## Regular Polygons

I. Section Objectives

• Recognize and use the terms involved in developing formulas for regular polygons.
• Calculate the area and perimeter of a regular polygon.
• Relate area and perimeter formulas for regular polygons to the limit process in prior lessons.

II. Cross- curricular-Architecture

• Use the following image of a roof in the shape of a hexagon.
• This is Figure 10.06.01
• www.space-frames.com/commercial buildings/xha28.htm
• Have the students use the dimensions of this design to figure out the area of the roof of this hexagon.
• Then have the students draw a diagram and explain how they figured out the area of the hexagon.
• Allow time for students to share their work when finished.

III. Technology Integration

• Have students complete some research on where to find hexagons and pentagons.
• Students can search architecture, nature or their own subject.
• Ask the students to keep track of the websites that they visit.
• Students should prepare a presentation of at least five examples of pentagons or hexagons in their given subject area.
• Students should include diagrams or images with their work.

IV. Notes on Assessment

• Assess student diagrams.
• How did the students figure out the area of the roof?
• Does their method make sense?
• Did they divide it into triangles?
• Did they divide it into trapezoids?
• Provide students with feedback on their work.

## Geometric Probability

I. Section Objectives

• Identify favorable outcomes and total outcomes.
• Express geometric situations in probability terms.
• Interpret probabilities in terms of lengths and areas.

II. Cross- curricular-Target Practice

• Use the following image of a dartboard.
• This is Figure 10.07.01.
• www.home.wlu.edu/~mcraea/GeometricProbabilityFolder/Introduction/Problem0/images/images/dartboard.gif
• Here is the problem.
• What is the geometric probability of hitting the center of the internal square of the dartboard?
• Use probability to figure this out.
• Allow time for students to share their work when finished.

III. Technology Integration

• Visit the same website that the image came from and explore the solution to the problem.
• www.home.wlu.edu/~mcraea/GeometricProbabilityFolder/Introduction/Problem0/images/images/dartboard.gif
• The answer to the problem that the student solved above is there.
• Have students use this to correct their own work.
• Show any changes/corrections that they completed.
• Then explore the other problems on the site.

IV. Notes on Assessment

• Because students are going to correct their own work during the technology integration, use this as a time to assess student work through observation.
• Are the students able to apply the concepts of probability to geometry?
• Refer students back to the text if they are having difficulty.

## Date Created:

Feb 22, 2012

Apr 29, 2014
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