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1.10: Perimeter and Area

Created by: CK-12

Triangles and Parallelograms

Pacing: This lesson should take one class period

Goal: This lesson introduces students to the area formulas for parallelograms and rectangles. It also illustrates the relationships between these formulas.

Flashcards! Creating another set of flashcards will be second-nature to our students by now. These flashcards should also be double-sided. The blank side should be a sketch of the figure and its special name. The flip side should repeat its definition, the sum of its interior angles, the expression for its perimeter, the formula for its area, and the formula for its perimeter. Have students create flashcards as the chapter presents the figures; this lesson only covers triangles, rectangles, and parallelograms.

Visualization! Encourage students to see how a parallelogram can be transformed into a rectangle by performing the activity presented in the lesson.

Extra Credit? You may want to offer extra credit for students who can correctly determine the total area of the eight circles found in the introduction of this lesson. 16 - 8*(\pi* (\frac{1}{2})^2) = 16-4 \pi \mathrm{ft}^2.

Extension! Using the hexagon below, find its area. Students use the concept of triangles and interior angles. Note** This may be more appropriate for students to attempt once the trapezoid area formula has been presented.

Trapezoids, Rhombi, and Kites

Pacing: This lesson should take one class period

Goal: This lesson further expands upon area formulas to include trapezoids, rhombi, and kites.

Additional Examples:

1. Trina has a rectangular flower garden, as shown below. The area of the garden is 1,602 \;\mathrm{ft}^2. How long is the bottom edge? Students must realize the bottom edge represents the altitude of the trapezoid.

2. What is the area of the kite pictured at the right?


3. Assume the rhombus below has an area of 45\;\mathrm{in}^2. One diagonal measures 5\;\mathrm{inches}. What is the length of the second diagonal? Extension: How long is each segment of the rhombus?

Areas of Similar Figures

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to connect similarity, areas, perimeters, and the scale factor k.

Look Out! Students may get confused when attempting to set up a proportion regarding similar area. Areas have a ratio of k^2, whereas perimeters have a ratio of k. This may be a good time to review the fraction \frac{\mathrm{image\ area}} {\mathrm{preimage\ area}} = \frac{k^2}{1}. Encourage students to use this proportion when referring to area.

Could There Be Giants? Before reading the section “Why There Are No 12’ Giants,” discuss Robert Wadlow, the tallest man on record. The following website is the “official” Robert Wadlow information guide. Students are captivated at Wadlow’s shoe size, his growth chart, and the photographs that can be found on this site. Encourage students to further research people of great height and do quick summaries of their findings.

Gulliver’s Travels! Written by Jonathan Swift, Gulliver’s Travels tells of a man who visits two worlds, one where people are 12\;\mathrm{times} his size (Brobdignagians), and another where people are \frac{1}{12} his size (Lilliputians). Use these values for k to describe such things as the area of footprints.


Circumference and Arc Length

Pacing: This lesson should take one to two class periods

Goal: The purpose of this lesson is to introduce the circumference formula and derive a formula for arc length (portions of the circumference)

Who Wants Pizza? Use pizza, pies, or cookies as a visual for this lesson’s formulas. Give each student one of the aforementioned round objects and a piece of string. Ask students to measure how much string it takes to circle around the object. Explain to students that this is the circumference.

When discussing arc length, split the object into six, eight, ten, or twelve even sections. Revisit central angles and the fraction of the whole. This fraction (\frac{1}{8}, \frac{1}{6}, \frac{1}{10}, \frac{1}{12}) will be your multiplier to the entire circumference.

Look Out! Students can become confused regarding the \mathrm{Pi} symbol (\pi). Students tend to view this as a variable instead of an approximate value.

Exact versus approximate. Students wonder why it is necessary to leave answer in exact value (\pi), instead of approximate (multiplying by 3.14). This is usually a teacher preference. By using the approximate value for \mathrm{Pi}, the answer automatically has a rounding error. Rounding the decimal too short will cause a much larger error than using the decimal to the hundred-thousandths place. Whatever your preference, be sure to explain both methods to your students.

Circles and Sectors

Pacing: This lesson should take one to two class periods

Goal: The purpose of this lesson is to introduce the area of a circle formula and derive a formula for its fractional area, the sector.

Arts and Crafts Time! Use a compass to draw a large circle. Fold the circle horizontally and vertically along its diameters and cut into four 90^\circ\;\mathrm{wedges}. Fold each wedge into quarters and cut along lines. Students should have 16\;\mathrm{wedges}. Fit all 16 pieces together to form a parallelogram, where the width of the parallelogram is the radius of the circle and the length is some value b. Students will see that the area of a sector must be a portion of the whole.

Who Wants Pizza? Use pizza, pies, or cookies as a visual for this lesson’s formulas. Give each student one of the aforementioned round objects. Illustrate area by discussing the amount of material needed to make the cookie, dough, etc. is an example of area. Have students discuss why the previously learned area formulas will not provide an accurate answer. Present the area of a circle formula and have students calculate the area of their individual object.

When discussing the area of a sector, divide the objects into 6, 8, 10, or 12 even sections. Each section represents a fraction of the whole, thus can be modeled by determining the fraction and multiplying it by the entire area.

Additional Examples:

  1. How much more pizza is in a 16” diameter pizza than a 12” diameter pizza? 87.96 \;\mathrm{in}^2
  2. Suppose a 14” pizza is cut into 10 slices. What is the area of two slices? 30.79\;\mathrm{in}^2

Regular Polygons

Pacing: This lesson should take one to two class periods

Goal: The purpose of this lesson is to introduce the formulas to determine the areas of regular polgyons by defining the apothem.

Additional Examples:

  1. Find the area of a regular pentagon with 11.8\;\mathrm{cm} sides and a 9.2\;\mathrm{cm} apothem. 271.4\;\mathrm{cm}^2.
  2. Find the area of a regular hexagon is its side length is 20”. 1039.23\;\mathrm{in}^2.
  3. Find the area of the regular quadrilateral below. 32\;\mathrm{mi}^2

Geometric Probability

Pacing: This lesson should take one class period

Goal: Students will apply the formula for general probability to geometric objects.

Extension! The probability formula can also be applied to areas. \mathrm{P} = \frac{\mathrm{area\ of\ favorable\ outcome}} {\mathrm{total\ area}}. Use this formula for the following additional examples.

Additional Examples:

1. What is the probability of landing in the bulls-eye of the dartboard below? \mathrm{Probability} = 1.56 \%

2. What is the probability that if you jump off the roof, you will land on the deck instead of the pool? \mathrm{Probability} = 73.63 \%

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