ACTIVITY: Deriving the Area Formula of a Trapezoid
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1 4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any trapezoid so tat its base lies on one of te orizontal lines of te paper. b. Estimate te area of your trapezoid (in square centimeters) by counting unit squares. b Area Estimate rigt angle c. Label te eigt and te bases inside te trapezoid. d. Cut out te trapezoid. Mark te midpoint of te side opposite te eigt. Draw a line from te midpoint to te opposite upper vertex. e. Cut along te line. You will end up wit a triangle and a quadrilateral. Arrange tese two figures to form a figure wose area you know cm b COMMON CORE Geometry In tis lesson, you will find areas of trapezoids. solve real-life problems. Learning Standard 6.G. f. Use your result to write a formula for te area of a trapezoid. Area = Formula g. Use your formula to find te area of your trapezoid (in square centimeters). Area = Exact Area b. Compare tis area wit your estimate in part (b). 66 Capter 4 Areas of Polygons
2 ACTIVITY: Writing a Mat Lesson Work wit a partner. Use your results from Activity to write a lesson on finding te area of a trapezoid. Mat Practice Use Clear Definitions Do your steps for te Key Idea elp anoter person understand ow to solve te problem? Do te examples follow your steps? Describe steps you can use to find te area of a trapezoid. Write two examples for finding te area of a trapezoid. Include a drawing for eac. Area of a Trapezoid Key Idea Use te following steps to find te area of a trapezoid Examples a. b. Write two exercises for finding te area of a trapezoid. Include an answer seet. Exercises Find te area IN YOUR OWN WORDS How can you derive a formula for te area of a trapezoid? 4. In tis capter, you used deductive reasoning to derive new area formulas from area formulas you ave already learned. Describe a real-life career in wic deductive reasoning is important. Use wat you learned about te areas of trapezoids to complete Exercises 4 6 on page 70. Section 4.3 Areas of Trapezoids 67
3 4.3 Lesson Lesson Tutorials Area of a Trapezoid Words Te area A of a trapezoid is one-alf te product of its eigt and te sum of its bases and b. Algebra A = ( + b ) b EXAMPLE Finding Areas of Trapezoids Find te area of eac trapezoid. a. 5 ft b. 8.5 m 6 ft 5 m 9 ft.5 m A = ( + b ) Write formula. A = ( + b ) (6)(5 + 9) Substitute. (5)( ) = (6)(4) Add. = (5)(0) = 4 Multiply. = 50 Te area of te trapezoid is 4 square feet. Te area of te trapezoid is 50 square meters. Exercises 7 9 Find te area of te trapezoid in. 8 mm 5 mm 4 mm 6 in..3 in. 68 Capter 4 Areas of Polygons
4 EXAMPLE Finding te Area of a Trapezoid on a Grid Wat is te area of te trapezoid? A 6 units B 7 units C 9 units D units Count grid lines to find te dimensions. Te eigt is 6 units, base is unit, and base b is units. A ( + b ) Write formula. (6)( + ) Substitute values. (6)(3) Add. = 9 Multiply. Te area of te trapezoid is 9 square units. Te correct answer is C. 6 EXAMPLE 3 Real-Life Application You can use a trapezoid to approximate te sape of Scott County, Virginia. Te population is about 3,00. About ow many people are tere per square mile? 0 mi 5 mi Find te area of Scott County. A ( + b ) Write formula for area of a trapezoid. = (0)(5 + 38) Substitute 0 for, 5 for, and 38 for b. (0)(53) = 530 Simplify. b 38 mi Te area of Scott County is about 530 square miles. Divide te population by te area to find te number of people per square mile. 3,00 people So, tere are about 44 people per square mile. 530 mi Exercises 3 3. Find te area of te trapezoid. 4. WHAT IF? In Example 3, te population of Scott County decreases by 550. By ow muc does te number of people per square mile cange? Explain. Section 4.3 Areas of Trapezoids 69
5 4.3 Exercises Help wit Homework. VOCABULARY Identify te bases and te eigt of te trapezoid.. REASONING Wat measures do you need to find te area of a trapezoid? 3. WHICH ONE DOESN T BELONG? Wic one does not belong wit te oter tree? Explain your reasoning. b w + w ( + b ) 4 ft 5 ft 7 ft 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-)= Find te area of te trapezoid. 4. = 4, b = 8, = 5. = 5, b = 7, = 4 6. =, b = 6, = in. 4 in cm cm 9. 0 ft 8 in. 7.5 ft 3.5 ft 3 cm 0. ERROR ANALYSIS Describe and correct te error in finding te area of te trapezoid. 6 m 8 m 4 m Area (6 + 4) = 0 m Find te area of te trapezoid LIGHT Ligt sines troug a window. Wat is te area of te trapezoid-saped region created by te ligt? 3 ft 4 ft 70 Capter 4 Areas of Polygons 5 ft
6 Find te area of a trapezoid wit eigt and bases and b. 5. = 6 in. 6. = cm 7. = mi 8. = 4 m = 9 in. = 0.5 cm = 5.6 mi = m b = in. b =.5 cm b = 7.4 mi b = m 9. REASONING Te rectangle and te trapezoid ave te same area. Wat is te lengt of te rectangle? 9 ft 9 ft 4 ft ft 0. OPEN-ENDED Te area of te trapezoidal student election sign is 5 square feet. Find two possible values for eac base lengt.. AUDIO How many times greater is te area of te floor covered ed by te larger speaker tan by te smaller speaker? b ft b. Te triangle and te trapezoid sare a 5-inc base and a eigt of 0 inces. a. Te area of te trapezoid is less tan twice te area of te triangle. Find te values of x. Explain your reasoning. b. Can te area of te trapezoid be exactly twice te area of te triangle? Explain your reasoning. x 5 in. 0 in. Plot te ordered pair in a coordinate plane. (Skills Review Handbook) 3. (5, 0) 4. (, 4) 5. (0, 3) 6. (6, ) 7. MULTIPLE CHOICE Wic expression represents 6 more tan x? (Section 3.) A 6 x B 6x C x + 6 D 6 x Section 4.3 Areas of Trapezoids 7
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