Math 8 Notes Unit 8: Area and Perimeter

Transcription

1 Math 8 Notes Unit 8: Area and Perimeter Syllabus Objective: (6.) The student will compute the perimeter and area of rectangles and parallelograms. Perimeter is defined as the distance around the outside of a figure. To find the perimeter of a figure, add the lengths of all its sides. Area is the number of square units in a figure. Finding the Perimeter of Rectangles and Parallelograms Example: Find the perimeter of the figure below. Since the length of all four sides is the same: P = sum of the sides P = P = 16cm 4 cm Example: Find the perimeter of the parallelogram below. Since it is a parallelogram, we know opposite sides are equal. 9 ft 5 ft P = P = P = 8 ft Example: Find the perimeter of the triangle below. We know the missing length is 15 feet. P = P = 4 ft 1' 15' Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 1 of 10 Revised 01 - CCSS

2 Finding the Area of a Rectangle One way to describe the size of a room is by naming its dimensions. A room that measures 1 feet by 10 feet would be described by saying it s a 1 by 10 foot room. In geometry, rather than talking about a room, we might talk about the size of a rectangular region. For instance, let s say I have a closet with dimensions ft by 6 ft. ft 6 ft Someone else might choose to describe the closet by determining how many one foot by one foot tiles it would take to cover the floor. To demonstrate, let me divide that closet into one foot squares. ft By simply counting the number of squares that fit inside that region, we find there are 1 squares. If I continue making rectangles of different dimensions, I would be able to describe their size by those dimensions, or I could mark off units and determine how many equally sized squares can be made. Rather than describing the rectangle by its dimensions or counting the number of squares to determine its size, we could multiply its dimensions together. 6 ft Putting this into perspective, we see the number of squares that fits inside a rectangular region is referred to as the area. A shortcut to determine that number of squares is to multiply the base by the height. The area of a rectangle is equal to the product of the length of the base and the length of the height. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page of 10 Revised 01 - CCSS

3 Example: Find the area of the rectangle. A = (9 cm)( cm) A = 18cm cm 9 cm Notice that the unit of measure, in this case centimeters, is squared. That happens as a result of multiplying centimeters times centimeters. That literally becomes centimeters squared (cm ). Finding the Area of a Parallelogram Have students explore and discover the formula for finding the area of a parallelogram. Using graph paper, have students draw a parallelogram. Have the students label the base and height. Now is a good time to remind students that the base and height of a polygon must meet at a right angle. Then have students cut out the parallelogram. Students can then cut a right triangle off the end of the parallelogram. By rearranging the triangle and fitting it on the other end of the parallelogram the students will create a rectangle. Therefore the formula for calculating the area of a parallelogram is the same and the formula for calculating the area of a rectangle. height base Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 3 of 10 Revised 01 - CCSS

4 Area of Rectangles and Parallelograms Words Numbers Formula 6 The area A of a rectangle or parallelogram is the base length b times the height h. A = (6)() = 1 units 6 A = (6)() = 1 units Syllabus Objective: (6.3) The student will describe how changes in the value of one variable effect the values of the remaining variables in a relationship. Once students have had the chance to practice finding the perimeter and area of rectangles and parallelograms, they should explore the effects of changing dimensions. Have students draw the following rectangles: 1 by, by 4, and 4 by 8. Then have them fill in the following table: Rectangle Base Height Perimeter Area 1 unit by units units by 4 units 4 units by 8 units Have the students use their table of values to answer the following question: How does changing the width and length of the rectangle affect the perimeter? The area? Make a conjecture about the perimeters and areas of similar figures. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 4 of 10 Revised 01 - CCSS

5 Using graph paper, draw rectangles with the following dimensions: 1 by units, by 4 units, and 4 by 8 units. Determine the perimeter and area for all 3 rectangles. P = P = 6 units A = 1 P = A = units P = 1 units A = 4 A = 8 units P = P = 4 units A = 48 A = 3 units When you double the base and the height, what happens to the perimeter? (doubles) Area? (quadruples) When you quadruple the base and the height, what happens to the perimeter? (quadruples) Area? (16 times) How might you show this relationship for area algebraically? if both base and height are doubled : A= b h A = 4bh The new area is 4 times the original area. The students should see that if the ratio that compares the perimeter of a large rectangle to that of a smaller similar rectangle is x, the ratio that compares the areas of the two rectangles will be x. You can have students continue to explore this by drawing more rectangles Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 5 of 10 Revised 01 - CCSS

6 Finding the Area of Triangles and Trapezoids Syllabus Objective: (6.4) The student will compute the area of triangles and trapezoids. A triangle or a trapezoid can be thought of as half of a parallelogram. You can clearly see that a triangle is simply half of a parallelogram. Therefore the area of a triangle is 1 Base Base 1 height To find the area of this parallelogram we would multiply the base which in this case is Base 1 + Base and the height. But a trapezoid is only half of the parallelogram so we end up with: Base 1 Base 1 A= ( b1+ b ) h Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 6 of 10 Revised 01 - CCSS

7 Area of Triangles and Trapezoids Words Triangle: The area A of a triangle is one-half 1 of the base length b times the height h. Trapezoid: The area of a trapezoid is one-half 1 the height h times the sum of the base lengths A= hb ( 1+ b ) b 1 and b. Formula Finding the Area and Circumference of a Circle Syllabus Objective: (6.5) The student will compute the area and circumference of circles. A circle is defined as all points in a plane that are equal distance (called the radius) from a fixed point (called the center of the circle). The distance across the circle, through the center, is called the diameter. Therefore, a diameter is twice the length of the radius, or d = r. We called the distance around a polygon the perimeter. The distance around a circle is called the circumference. There is a special relationship between the circumference and the diameter of a circle. Let s get a visual to approximate that relationship. Take a can with 3 tennis balls in it. Wrap a string around the can to approximate the circumference of a tennis ball. Then compare that measurement with the height of the can (which represents three diameters). You will discover that the circumference of the can is greater than the three diameters (height of the can). You can make an exercise for students to discover an approximation for this circumference/diameter relationship which we call π. Have students take several circular objects, measure the circumference (C) and the diameter (d). Have students determine C for each object; d have groups average their results. Again, they should arrive at answers a little bigger than 3. This should help convince students that this ratio will be the same for every circle. C We can then introduce that = π or C = πd. Since d = r, we can also write C = π r. Please d note that π is an irrational number (never ends or repeats). Mathematicians use π to represent the exact value of the circumference/diameter ratio. Example: If a circle has a diameter of 4 m, what is the circumference? Use 3.14 to approximate π. State your answer to the nearest 0.1 meter. C = π d Using the formula: C (3.14)(4) C 1.56 The circumference is about 1.6 meters. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 7 of 10 Revised 01 - CCSS

8 Many standardized tests (including the CRT and the district common exams) ask students to leave their answers in terms ofπ. Be sure to practice this. Example: If a circle has a radius of 5 feet, find its circumference. Do not use an approximation forπ. C = π r Using the formula: C = π 5 C = 10π The circumference is about 10π feet. Example: If a circle has a circumference of 1π inches, what is the radius? Using the formula: C = π r 1π = πr 6π = πr 6 = r The radius is 6 inches. Circumference of a Circle Words Numbers Formula The circumference C of a C = π d C = π d circle is π times the diameter C = π (6) OR d, or π times the radius r. C = ( π )(3) 3 C = π r C You can demonstrate the formula for finding the area of a circle. First, draw a circle; cut it out. Fold it in half; fold in half again. Fold in half two more times, creating 16 wedges when you unfold the circle. Cut along theses folds. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 8 of 10 Revised 01 - CCSS

9 Rearrange the wedges, alternating the pieces tip up and down to look like a parallelogram. You will notice that the length is equal to ½ of the distance around the circle. The height is equal to the r of the circle. radius (r) This is ½ of the distance around the circle or ½ of C. We know that C = π r, so 1 1 C = π r 1 C = πr The more wedges we cut, the closer it would approach the shape of a parallelogram. No area has been lost (or gained). Our parallelogram has a base of π r and a height of r. We know from our previous discussion that the area of a parallelogram is bh. So we now have the area of a circle: radius (r) A= (π r)( r) A= πr πr Example: find the area of the circle to the nearest square meter if the radius of the circle is 1 m. Useπ Using the formula: A= π r A (3.14)(1) A The area is about 45 square meters. Example: If the area of a circle is 70 square meters, find the radius to the nearest meter. Using the formula: A= π r 70 (3.14) r.3 r.3 r 4.7 r The radius of the circle is about 5 meters. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 9 of 10 Revised 01 - CCSS

10 Area of a Circle Words Numbers Formula The area A of a circle is π A= π r A= π r times the square of the radius r. A = π (3 ) A = 9π 3 Students should also practice finding the area of irregular figures by breaking it up into familiar figures. Example: The dimensions of a church window are shown below. Find the area of the window to the nearest square foot. First, find the area of the rectangle. A = 11 8 A = 88 Next, we have half of a circle. We are given the diameter, so the radius would be half of the 11 feet or 5.5 feet. To find the area of half of a circle with radius 5.5, 8 feet 1 A= π r 1 A (3.14)(5.5) A feet To find the total area we add the two areas we found: The area of the church window is about 136 square feet. Holt: Chapter 8, Sections 1-3 Math 8, Unit 8: Area and Perimeter Page 10 of 10 Revised 01 - CCSS