Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make is a rectangle. Anoter is parallelogram a four-sided figure wit opposite sides parallel and equal in lengt a parallelogram. How do you calculate te area of te rectangle? Can you use te same metod to calculate te area of te parallelogram? How do you determine te area of a parallelogram? 1. centimetre grid paper ruler scissors tape 100 MHR Capter 3 On centimetre grid paper, draw a rectangle tat is 6 cm long and 4 cm wide. Cut out te rectangle wit scissors.
2. Count te number of square centimetres te rectangle covers. Wat is te area of tis rectangle? 3. Use scissors to cut across te rectangle as sown. Tape te two pieces togeter. 4. Wat sape did you form? Wat do you know about tis sape tat elped you to identify it? 5. Is te area of te parallelogram te same as tat of te original rectangle? How do you know? 6. a) Predict te lengt of te base (b) of te parallelogram. Verify by measuring wit a ruler. b) Predict te eigt () of te parallelogram. Verify by measuring wit a ruler. c) Is parallel or perpendicular to b of te parallelogram? 7. Wat is te relationsip between b and, and te area of te parallelogram? Reflect on Your Findings 8. a) Suggest a formula for calculating te area of a parallelogram. b) Compare your formula wit tose of your classmates. Discuss any differences and make sure tat everyone agrees on te formula. base a side of a two-dimensional closed figure common symbol is b eigt te perpendicular distance from te base to te opposite side common symbol is b 3.4 Area of a Parallelogram MHR 101
Example 1: Determine te Area of a Parallelogram Te parallelogram sown as a base of 5 cm and a eigt of 3 cm. Use a formula to determine its area. 3 cm Solution Base (b) is 5 cm and eigt () is 3 cm. Substitute te values into te formula for te area of a parallelogram. A = base eigt A = b A = 5 3 A = 15 Te area of te parallelogram is 15 c m 2. Ceck: Verify your answer. Estimate te area by counting squares. Count full squares: 12 squares. Count squares tat are almost full: 2 squares. Count squares tat are about alf full: 2 alf squares = 1 full square. Do not count almost empty squares. 12 + 2 + 1 = 15 Te estimate for te area of te parallelogram is 15 cm 2. 5 cm Tis parallelogram was drawn on centimetre grid paper. Use a formula to determine its area. Ceck your answer using estimation. Sometimes it is necessary to extend te line of te base to measure te eigt. 102 MHR Capter 3
Example 2: Make Calculations Using te Area of a Parallelogram Jessica as created a tulip garden in te sape of a parallelogram. Se is going to plant 10 tulips per square metre of garden. Jessica s garden as a base of 8 m and a eigt of 5 m. How many tulips does Jessica need? 5 m 8 m Solution Wat is te area of te garden? How many tulips does Jessica need? 1. Determine te area of te garden. Understand Plan 2. Determine te number of tulips needed for te area. 1. To find te area of te garden, substitute te values into te formula. A = b A = 8 5 A = 40 Te area of te garden is 40 m 2. 2. Jessica is going to plant 10 tulips per square metre. Tat means eac square metre of er garden will ave 10 tulips in it. 1 m 2 10 tulips 2 m 2 20 tulips 3 m 2 30 tulips Te pattern is to multiply te number of square metres by 10 to get te number of tulips. Do It! Strategies Look for a Pattern Refer to page xvii. 40 10 = 400 Jessica needs 400 tulips. Ceck: Look Back Work backward by dividing 400 tulips by 10. Te answer sould be te area of Jessica s garden. 400 10 = 40. Te calculated area for Jessica s garden was 40 m 2. 3.4 Area of a Parallelogram MHR 103
Example 3: Determine te Heigt of a Parallelogram Diana as enoug parallelogram-saped tiles to cover a section of wall wit an area of 840 c m 2. Te design will ave a base of 12 cm. How ig can se make er design? Strategies Guess and Ceck Refer to page xvi. Solution Determine te eigt (). Te base b is 12 cm and te area is 840 c m 2. Substitute tese values into te formula for te area of a parallelogram. A = b 840 = 12 Guess and ceck to determine. Try = 50 12 50 = 600 Too low. Try = 80 12 80 = 960 Too ig. Try = 70 12 70 = 840 Correct! Diana can make te eigt of te design 70 cm. b = 12 cm A parallelogram as a base of 90 m and an area of 450 m 2. Determine te eigt of te parallelogram. Te formula for te area of a rectangle can be used to determine te formula for te area of a parallelogram. Te formula for te area of a parallelogram is A = b, were b is te base and is te eigt. Te eigt of a parallelogram is always perpendicular to its base. b A = b b A = b 1. Explain wy te area of a rectangle and te area of a parallelogram made from te rectangle are te same. Include diagrams wit your answer. 2. Deepa determined te area of tis parallelogram to be 5 c m 2. a) Explain er error. b) Do te calculation correctly. 2 cm 2.5 cm 104 MHR Capter 3
For elp wit #3 to #6, refer to Example 1 on page 102. 3. Tese parallelograms are drawn on centimetre grid paper. Wat is te area of eac parallelogram? a) b) 6. Draw eac parallelogram on centimetre grid paper. Determine eac area using te formula. Ceck your answers using estimation. a) = 6 cm, b = 4 cm b) = 2 cm, b = 4 cm For elp wit #7 and #8, refer to Example 2 on page 103. 7. Wat is te area of eac parallelogram? a) 5 cm 11 cm b) b = 7 cm, = 9 cm c) Te eigt is 6.2 m and te base is 3 m. 4. Determine te area of eac parallelogram. Tey are drawn on centimetre grid paper. a) 8. Determine te area of eac parallelogram. a) 9 m b) 3 m b) b = 9.2 mm, = 12.3 mm c) Te base is 4.5 cm and te eigt is twice te base. 5. Draw eac of te following parallelograms on centimetre grid paper. Use te formula to determine te area of eac parallelogram. Ceck your answers using estimation. a) b = 4 cm, = 5 cm b) b = 3 cm, = 7 cm For elp wit #9 and #10, refer to Example 3 on page 104. 9. Workers are putting new flooring in te community centre. A section of te floor is going to ave a parallelogramsaped design. Tere are enoug floor tiles b = 3 m to cover an area of 12 m 2. Te base will be 3 m. Wat can te eigt of te design be? 3.4 Area of a Parallelogram MHR 105
10. A parallelogram-saped stripe is going to be painted on te wall of a daycare centre. Tere is enoug paint to make a stripe wit an area of 9 m 2. How ig can te stripe be? 13. Te spring taw as caused a rectangular section of te scoolyard to flood. A parallelogram-saped wooden walkway is being placed over it as sown. Te area of te walkway is 22 m 2. How wide will te base of te walkway be in metres? = 11 m b = 1.5 m b 11. a) Wit a ruler, measure te base and eigt of tis parallelogram. Explain ow you measured. 14. Wanda as designed a logo of er first initial on centimetre grid paper. Se wants to find te area of te logo so se can get fabric to sew it on er jacket. Wat is te area of te initial? b) Wat is te area? c) Use a different side of te parallelogram as te base. Determine te new eigt and calculate te area. How does tis compare to your answer in b)? 12. Wic of te two sapes sown below as a greater area? Explain your answer. 15. Wat is te area of parallelogram ABCD? y A B 4 Sape A Sape B 2 D C 2 0 2 4 6 x 6 cm 2 cm 6 cm 2 cm 16. Wat is te area of te yellow portion of tis corporal s insignia? Use a calculator to determine te answer. 6 cm 0.9 cm 106 MHR Capter 3
17. A construction company is building a road troug a rectangular section of farmland. Te farmer wants to know te area of land being used for te road. Determine te area in square metres. 19. Te figure below is a parallelogram. Wat is te lengt of n? 15 m 1.8 km 8 cm n 10 cm 5.4 km 12 cm How many metres are in one kilometre? 18. a) Wat is te area of paving stone needed for te patio? 20. Wat is te area of tis X? Explain ow you calculated your answer. 10 m 6 m 9.5 m b) Paving stones cost $55 per square metre. Estimate ow muc it would cost to pave te patio. Sow your work. c) Calculate te cost. MATH LINK a) You are going to include in your airport design an area tat is in te sape of a parallelogram. Add as many runways and taxi lanes as necessary to create a parallelogram. b) Determine te area of te parallelogram. If you ave more tan one parallelogram, use te largest one. c) You decide to plant grass in te parallelogram-saped area. If one bag of grass seed covers 1000 m 2, ow many bags of seed will you need for te parallelogram? 3.4 Area of a Parallelogram MHR 107