New Jersey Center for Teaching and Learning Slide 1 / 183 Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org Slide 2 / 183 Perimeter and Area March 1, 2012 www.njctl.org Table of Contents Perimeter Circumference Area of a Square Area of a Rectangle Area of a Rectangular Region Area of an Irregular Region Area of a Triangle Area of a Parallelogram Area of Kites and Rhombi Area of a Trapezoid Area of a Regular Polygon Area of a Circle Area of Sectors and Segments of a Circle Heron's Formula Brahmagupta's Formula Geometric Probability Slide 3 / 183
Slide 4 / 183 Perimeter Return to Table of Contents Perimeter is the linear distance (ie. ft, cm, m) around the boundary of a plane figure. Slide 5 / 183 14 in 19 in 15 in 6 in 19 in 6 in 15 in + 14 in 54 in Common Perimeter Formulas Slide 6 / 183 Rectangle: Square: Regular Polygon:, where n is the number of sides of length s.
1 Find the perimeter of the figure (in feet.) Slide 7 / 183 4 ft 8 ft 6 ft 10 ft 9 ft 2 Find the perimeter of the figure (in cm). Slide 8 / 183 12 cm 9 cm 5 cm 4 cm 7 cm 3 Find the perimeter of the figure. 10 cm 3 cm 3 cm 7 cm 3 cm 2 cm 3 cm Slide 9 / 183 4 cm
4 Find the perimeter of the figure. Slide 10 / 183 30 in 4 ft 2 ft 5 ft 5 The perimeter of a square is 30 in, what is the length of a side? Slide 11 / 183 6 The perimeter of a rectangle is 18 ft and has length 8 ft, what is the width? Slide 12 / 183
Slide 13 / 183 Circumference Return to Table of Contents Slide 14 / 183 # is the constant ratio of circumference to diameter. Slide 15 / 183 #, pi, is a non-terminating, non-repeating real number, which we call an irrational. For problems that ask for an exact value, leave # in the answer. For problems that do not specify exact value 3.14, 22/7, or the # button on your calculator may be used.
Example: Find the circumference of circle A. Slide 16 / 183 8 ft A Example: Find the circumference of circle B. B 6 in Example: The circumference of a circle is 6# m, find the diameter. Slide 17 / 183 The diameter is 6m. Example: The circumference of a circle is 12 in, find the radius. 7 What is the circumference of a circle with radius 6? (use ) Slide 18 / 183
8 What is the circumference of a circle with diameter 9? (use ) Slide 19 / 183 9 What is the diameter of a circle with circumference 10#? (use ) Slide 20 / 183 10 What is the radius of a circle with circumference 16? (use ) Slide 21 / 183
11 What is the perimeter of the figure? (use ) Slide 22 / 183 6 4 12 What is the perimeter of the figure? (use ) Slide 23 / 183 6 13 What is the perimeter of the figure? (use ) Slide 24 / 183 5
Arc Length of a circle is part of the circumference. Slide 25 / 183 Using this ratio will take a fraction of the circumference. Can use either 2# r or d#. Example: Find the length of an arc of a circle with radius 6 and central angle 40 o. Slide 26 / 183 Example: The length of an arc of a circle, with diameter 8, is #. Find the angle measure. Slide 27 / 183
14 Find the length of the arc of a circle with radius 7 and arc measure 45 o. Slide 28 / 183 15 Find the length of the arc of a circle with circumference 20# and central angle measure 90 o. Slide 29 / 183 16 The length of the arc of a circle with, circumference 18#, is 6#. Find the diameter. Slide 30 / 183
17 The length of the arc of a circle with, circumference 30#, is 4#. Find the area of the circle. Slide 31 / 183 Slide 32 / 183 Area of a Square Return to Table of Contents Area is a two-dimensional measure, as oppose to perimeter which was length. Slide 33 / 183 The area of a figure represents the number of congruent squares that fit inside of the figure. 4 in 4 in A 4 inch square will hold 16 square inches, or 16 in 2.
Area of a Square = side length 2 Slide 34 / 183 Example: Find the area of a square with sides 7ft. The area is 49 ft 2. Example: Find the length of a side of a square with area 36 m 2. The sides have a length of 6 m. Example: Find the area of a square with perimeter of 20 ft. Slide 35 / 183 The area of the square is 25 ft 2. Example: Find the perimeter of a square with area 64 cm 2. Slide 36 / 183 The perimeter is 32 cm.
18 Find the area of the square with sides 2ft. Slide 37 / 183 19 Find the area of the square with sides 11 in. Slide 38 / 183 20 Find the area of the square with sides 1/2 mm. Slide 39 / 183
21 Find the area of the square with perimeter of 20 cm. Slide 40 / 183 22 Find the side of a square with area 81 yd 2. Slide 41 / 183 23 Find the perimeter of a square with area 100 m 2. Slide 42 / 183
24 How many square feet are in a square that is 1 yard by 1 yard? Slide 43 / 183 25 How many square inches are in a square that is 1 foot by 1 foot? Slide 44 / 183 Slide 45 / 183 Area of a Rectangle Return to Table of Contents
The area of a rectangle = length x width Slide 46 / 183 8 ft 3 ft Slide 47 / 183 Example: Find the area of a rectangle that is 6 cm by 8 cm. Example: Find the width of a rectangle with area of 48 yd 2 and a length of 12 yd? Example: Find the area of a rectangle with length of 6 in and width 3 ft. Slide 48 / 183 Example: The width of a rectangle is 7 cm and the perimeter is 38 cm. Find the area of the rectangle.
26 Find the area of the rectangle. Slide 49 / 183 8ft 2ft 27 Find the area of the rectangle. Slide 50 / 183 16in 2ft 28 Find the area of the rectangle. Slide 51 / 183 16in 2ft
29 Find the length of the rectangle. Slide 52 / 183 A=96 sq ft 12ft 30 Find the perimeter of the rectangle. Slide 53 / 183 10 in A= 90 sq in Slide 54 / 183 Area of a Rectangular Region Return to Table of Contents
The area of a figure can be found by splitting into known regions, such as rectangles. 5 cm 12 cm I 9 cm 5 cm 4 cm II 7 cm Area of Region I= 5(12)=60 cm 2 Area of Region II= 4(7)=28 cm 2 Total Area= 60 + 28= 88cm 2 Slide 55 / 183 Suppose instead of vertical rectangles, we use horizontal? 12 cm 5 cm I II 9 cm 5 cm 4 cm 7 cm Area of Region I= 5(5)=25 cm 2 Area of Region II= 9(7)=63 cm 2 Total Area= 25 + 63= 88cm 2 Slide 56 / 183 The area is the same! So a figure can be cut into whatever rectangles you want as long as there is no overlap. Example: Find the area. 10 cm 3 cm 7 cm 3 cm 2 cm 3 cm 3 cm 4 cm Slide 57 / 183
Another way to find the area is to see what the larger rectangle it was created from and subtract the missing parts. 5 cm 5 cm Slide 58 / 183 12 cm 9 cm 4 cm 7 cm 12(9) - 5(4) = 108-20= 88 cm 2 Example: Find the area of the figure. Slide 59 / 183 2in 4in 6in 8in 8in 31 Find the area of the region. Slide 60 / 183 4 6 3 5 2 12
Slide 61 / 183 32 Find the area of the region. 8 6 16 11 10 30 33 Find the area of the region. Slide 62 / 183 3 1 3 4 8 3 3 6 34 Find the area of the shaded region. Slide 63 / 183 15 4 1 4
Slide 64 / 183 Area of an Irregular Region Return to Table of Contents How would we find the area of an irregular region? Slide 65 / 183 Would dividing it into rectangles make sense? What about dividing it into squares? We can find the area by overlaying a grid and then counting the number of complete and partial squares that region fills. Slide 66 / 183 There are 2 completely full squares and 15 partially filled squares. Some of the partially have very little of the region, others are almost completely covered, so on average they make about 7.5 full squares, for a total of 9.5 squares. Since each square is a square inch, the approximate area is about 9.5 sq in.
The approximate Area of an Irregular Region Slide 67 / 183 F= Number of full squares P= Number of partially filled squares A= Area of one square 35 How many completely filled squares are there? Slide 68 / 183 36 How many completely partially filled squares are there? Slide 69 / 183
37 What is the area of one square? Slide 70 / 183 38 What is the approximate area of the region? Slide 71 / 183 39 What is the approximate area of the region? Slide 72 / 183
How can we improve the accuracy of our approximation? Slide 73 / 183 1in Slide 74 / 183 Slide 75 / 183 So the smaller the square used the better the approximation. In mathematics, we describe this as the size approaches zero the approximation approaches the actual area.
Slide 76 / 183 Area of a Triangle Return to Table of Contents Recall the area of a triangle is Slide 77 / 183 The height is the distance, measured along the perpendicular, from a vertex to the line containing the opposite side. h h h b b b Notice the implied definition of base. It is not necessarily the "bottom" but the side to which the height is drawn. Slide 78 / 183 Identify the base. h h h
40 Find the area of the triangle Slide 79 / 183 4 8 41 Find the area of the triangle Slide 80 / 183 5 3 7 42 Find the area of the triangle Slide 81 / 183 8
43 Find the area of the triangle Slide 82 / 183 17 8 10 44 Find the area of the figure Slide 83 / 183 10 5 8 45 Find the value of h. Slide 84 / 183 5 h 12 8
46 Which triangle has the greatest area? Slide 85 / 183 A A B C D B C D E They are all equal Slide 86 / 183 Area of a Parallelogram Return to Table of Contents When looking at a shape that we don't the formula for area for, try to make into shapes you know the formula for. 12 8 9 If the parallelogram is cut along the height. Slide 87 / 183 8 12 9 Move the triangle to opposite side. 8 12 9 The 2 shapes make a rectangle. 8 12 9 What is the area of the rectangle? What is the area of the original parallelogram? How can we find the area of a parallelogram?
Area of a Parallelogram Slide 88 / 183 4 10 10 4 Example: Find the area of the parallelograms. Slide 89 / 183 4 3 8 5 7 6 Slide 90 / 183
47 Find the area of the paralelogram. Slide 91 / 183 5 4 12 48 Find the area of the paralelogram. Slide 92 / 183 7 10 3 49 Find the area of the paralelogram. Slide 93 / 183 10 8 14
Slide 94 / 183 Slide 95 / 183 52 Find the value of x. Slide 96 / 183 5 x 12 8
Slide 97 / 183 Area of Kites and Rhombi Return to Table of Contents When looking at a shape that we don't the formula for area for, try to make into shapes you know the formula for. Slide 98 / 183 1 / 2d 2 1 / 2d 2 d 1 d 2 d 1 The diagonal connecting the vertices included by the congruent sides, longer one, bisects the other diagonal. Cut along the longer diagonal. The area of each triangle is 1 / 2d 1( 1 / 2d 2)= 1 / 4d 1d 2 The area of the kite is 2( 1 / 4d 1d 2)= 1 / 2d 1d 2 The Area Formula for Kites and Rhombi Slide 99 / 183 (Since a square is a kite, its area can be found using the same formula.)
Example: Find the area Slide 100 / 183 4 3 4 17 8 17 5 10 10 Example: Find the area of the square. Slide 101 / 183 10 53 Find the area of the figure. Slide 102 / 183 5 4 6 4
54 Find the area of the kite. Slide 103 / 183 5 4 3 55 Find the area of the rhombus. Slide 104 / 183 3 5 56 Find the area of the figure. Slide 105 / 183
57 Find the area of the figure. Slide 106 / 183 Slide 107 / 183 Area of a Trapezoid Return to Table of Contents The median of a trapezoid is the segment that connects the midpoints of the non-parallel sides. Slide 108 / 183 base 1 median base 2
58 Find the value of x. Slide 109 / 183 20 8 x 59 Find the value of x. Slide 110 / 183 4 x 24 60 Find the value of x. Slide 111 / 183 x 12 16
When looking at a shape that we don't the formula for area for, try to make into shapes you know the formula for. Slide 112 / 183 In terms of the original trapezoid, what is the area of this figure? Area of a Trapezoid Slide 113 / 183 Example: Find the area. 8 Slide 114 / 183 3 8 5 14
Slide 115 / 183 Example: Find the area. 2 5 6 8 1 4 3 61 Find the area of the trapezoid. Slide 116 / 183 10 6 14 62 Find the area of the trapezoid. Slide 117 / 183 4 3 3 8
Slide 118 / 183 Slide 119 / 183 65 The area of a trapezoid is 80sq ft. If the bases are 16 ft and 24 ft, how many feet long is the height? Slide 120 / 183
66 The area of a trapezoid is 48 sq cm. If the height is 6 cm and one base are 12 cm, how many centimeters long is the other base? Slide 121 / 183 Slide 122 / 183 Area of a Regular Polygon Return to Table of Contents Parts of a Regular Polygon Slide 123 / 183 side r a radius center apothem 1 / 2s Recall the formula for finding the measure of 1 interior angle
When looking at a shape that we don't the formula for area for, try to make into shapes you know the formula for. Slide 124 / 183 a s The regular hexagon can be split into 6 congruent triangles. The area of a triangle is in terms of the hexagon The area of the hexagon would be and associative properties of multiplication What does 6s represent in the hexagon. using the commutative The Area of a Regular Polygon Slide 125 / 183 a = apothem of the regular polygon p = perimeter of the regular polygon Slide 126 / 183 Example: Find the area of the regular polygon. 4 4 3 5
67 Find the area of the regular polygon. Slide 127 / 183 6 5 68 Find the area of the regular polygon. Slide 128 / 183 6 8 Slide 129 / 183 Example: Find the area of an equilateral triangle with apothem 6 6 Example: Find the area of an equilateral triangle with sides 6 6
Example: Find the area a regular pentagon with perimeter 40 in. Slide 130 / 183 69 Find the area of the regular polygon. Slide 131 / 183 4 70 Find the area of the regular polygon. Slide 132 / 183 9
71 Find the area of the regular polygon. Slide 133 / 183 12 Example: Find the area of the shaded region created by these 2 regu polygons. Slide 134 / 183 The span is 16 7 72 Find the area of the shaded region. Slide 135 / 183 16
73 Find the area of the shaded region. Slide 136 / 183 8 Slide 137 / 183 Area of a Circle Return to Table of Contents Area Formula for a Circle Slide 138 / 183 Example: Find the exact area of a circle with radius 6 in. Using 3.14 or the pi key on your calculator only gives an approximate value. Why? in 2
74 What is the area of a circle with radius 8? (use 3.14 for pi) Slide 139 / 183 75 What is the area of a circle with diameter 10? (use 3.14 for pi) Slide 140 / 183 76 What is the area of a circle with circumference 20#? (use 3.14 for pi) Slide 141 / 183
77 What is the radius of a circle with area 18# u 2? (use 3.14 for pi) Slide 142 / 183 78 What is the circumference of a circle with area 40 u 2? (use 3.14 for pi) Slide 143 / 183 Slide 144 / 183 Area of Sectors and Segments of a Circle Return to Table of Contents
Area of a Sector A sector is a "slice" of a circle. Slide 145 / 183 r x o How much of the circle the sector represents. Area of the circle Example: Find the area of the sector. Slide 146 / 183 9 120 o The area of the sector is 27# units 2. 79 Find the area of the sector. (use 3.14 for pi) Slide 147 / 183 5 100 o
80 Find the area of the sector. (use 3.14 for pi) Slide 148 / 183 d=12 81 Find the area of the figure. (use 3.14 for pi) Slide 149 / 183 diameter 10 8 82 If the area of a sector of a circle with radius 10 is 20, find the arc measure of the sector. (use 3.14 for pi) Slide 150 / 183
83 If the area of a sector of a circle is 30#, find the arc length of the sector if the arc measure is 40 o. (use 3.14 for pi) Slide 151 / 183 Area of a Segment of a Circle Slide 152 / 183 A segment of a circle is a sector with the isosceles triangle, with legs that are radii and a base that is a chord, removed. Slide 153 / 183
Slide 154 / 183 Slide 155 / 183 Slide 156 / 183 Heron's Formula Return to Table of Contents
Slide 157 / 183 In Alexandria, Roman Egypt, about 10 AD, a mathematician named Heron (also pronounced Hero) was born. He derived a formula based on the sides of a triangle. In his formula he used a measure called a semi-perimeter. semi-perimeter = 1 / 2 perimeter Heron's Formula Slide 158 / 183 a, b, & c are the lengths of the sides of the triangle Example: Find the area of a triangle with sides 5, 6, 7. Slide 159 / 183
86 Find the area of the triangle with sides 6, 7, and 8. Slide 160 / 183 87 Find the area of the triangle with sides 5, 5, and 8. Slide 161 / 183 88 Find the area of the triangle with sides 3, 6, and 10. Slide 162 / 183
Slide 163 / 183 Brahmagupta's Formula Return to Table of Contents Slide 164 / 183 The Indian mathematician Brahmagupta was born in 598 AD. He wrote a mathematics book that contained several mathematical theorems. The most famous of these, was a formula for calculating the area of a quadrilateral that was cyclic (the four vertices lie on the same circle.) Slide 165 / 183
Example: Find the area of ABCD Slide 166 / 183 A 8 6 7 3 B C D 89 Find the area of JKLM. Slide 167 / 183 J 6 M 10 5 K 9 L 90 Find the area of JKLM. Slide 168 / 183 7 10 4 3
91 Find the area of JKLM. Slide 169 / 183 9 12 13 center Slide 170 / 183 Geometric Probability Return to Table of Contents Probability of an Event Slide 171 / 183 Probability of rolling a 5: Probability of picking a club from a deck of cards
Geometric Probabilities Slide 172 / 183 A commuter train breaks down on a 40-mile route what is the probability that it occurs between 2 stations that are 6 miles apart? 92 A toy train is a 6 ft loop, if the train stops randomly what is the probability that the front of the locomotive is within 6 inches of the station? (enter as a fraction) Slide 173 / 183 Geometric Probabilities Slide 174 / 183 Sector A has measure 180 o. Sector B has measure 90 o. Sector C has measure 30 o. Sector D has measure 60 o. B What are the following: P(A)= 1/2 P(C)= 1/12 P(B or D)= 5/12 P(~D)= 5/6 C D A
93 Find P(A) (enter answer as fraction) Slide 175 / 183 B 120 o A 94 Find P(D) (enter answer as fraction) C Slide 176 / 183 B 20 o D A 95 Find P(C or A) (enter answer as fraction) C Slide 177 / 183 B 20 o D A
96 Find P(~B) (enter answer as fraction) C Slide 178 / 183 B 20 o D A Geometric Probabilities Slide 179 / 183 What is the probability that a point inside the circle is also inside the square? Area of Circle Area of Square 6 Geometric Probabilities Slide 180 / 183 What is the probability that a point inside the circle is not inside the square? Area of Circle Area of Square 6 Area of Shaded
97 What is the probability of a point inside the triangle is inside the circle? (Enter answer in decimal form) Slide 181 / 183 10 4 10 98 What is the probability of a point inside the regular hexagon with apothem 4 does not lie inside the triangle with radius 4? (Enter answer in decimal form) Slide 182 / 183 99 If a randomly thrown dart hits a dart board with a 12" diameter, what is the probability that it hits the bulls eye that has a diameter of 1"? (Enter answer in decimal form) Slide 183 / 183