8.5 Area of Parallelograms
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1 8. Area of Parallelograms Goal Find te area of parallelograms. Key Words ase of a parallelogram eigt of a parallelogram parallelogram p. 0 romus p. Geo-Activity Use a straigtedge to draw a line troug one of te vertices of an inde card. Eploring te Area of a Parallelogram Cut out te triangle. Tape te triangle to te opposite side to form a parallelogram. How does te area of te parallelogram compare to te area of te rectangular inde card? How do teir ases compare? How do teir eigts compare? Write a conjecture aout te formula for te area of a parallelogram. Eiter pair of parallel sides of a parallelogram are called te ases of a parallelogram. Te sortest distance etween te ases of a parallelogram is called te eigt of a parallelogram. Te segment tat represents te eigt is perpendicular to te ases. eigt ase A eigt can e inside te parallelogram. ase A eigt can e outside te parallelogram. As te Geo-Activity suggests, te area of a parallelogram is found using a ase and its corresponding eigt. AREA OF A PARALLELOGRAM Words Area (ase)(eigt) Symols A 8. Area of Parallelograms 9
2 EXAMPLE Find te Area of a Parallelogram Find te area of te parallelogram. m Use te formula for te area of a parallelogram. Sustitute 9 for and for. 9 m A Formula for te area of a parallelogram (9)() Sustitute 9 for and for. Multiply. ANSWER Te parallelogram as an area of square meters. EXAMPLE Find te Heigt of a Parallelogram Find te eigt of te parallelogram given tat its area is 78 square feet. A 78 ft A Formula for te area of a parallelogram 78 Sustitute 78 for A and for.. Divide eac side y. ft ANSWER Te parallelogram as a eigt of. feet. Area of Parallelograms Find te area of te parallelogram mm yd mm 8 yd ft ft In Eercises, A gives te area of te parallelogram. Find te missing measure.. A 7 in.. A 0 m. A 8 cm in. m 7 cm 0 Capter 8 Polygons and Area
3 AREA OF A RHOMBUS Words Area (product of diagonals) Symols A d d d d Area of a Romus Te formula for te area of a romus can e justified using te area of a triangle. A specific case is given elow. Te diagonals divide a romus into four congruent rigt triangles. So, te area of te romus is times te area of one of te rigt triangles. Area of triangle ()() square units 8 Area of triangles () square units Notice tat d d, or ()(8), also equals square units. IStudent Help I CLASSZONE.COM MORE EXAMPLES More eamples at classzone.com EXAMPLE Find te Area of a Romus Find te area of te romus. a.. in. m 9 m 9 m m 0 in. a. A d d. A d d ()(0) ( )(9 9) 70 ()(8) 08 Te area of te romus is 70 square inces. Te area of te romus is 08 square meters. 8. Area of Parallelograms
4 8. Eercises Guided Practice Vocaulary Ceck Skill Ceck. Te parallelogram at te rigt as a ase of 7 units and a eigt of units. Sketc te parallelogram and lael its ase and eigt. Matc te quadrilateral wit te corresponding area equation. A. A ()(0) B. A ()() C. A (0)()... 0 Find te measures of te diagonals of te romus m 8 ft m m ft m cm cm cm cm Practice and Applications Etra Practice See p. 90. Area of Parallelograms Find te area of te parallelogram Hom work H lp Eample : Es. 8 Eample : Es. 8 Eample : Es. 8 You e te Judge. One of your classmates states tat it is possile to draw a parallelogram tat is different from te one at te rigt, ut as te same area. Do you agree or disagree? Eplain your reasoning. ft 7 ft Capter 8 Polygons and Area
5 Crafts. Stained Glass Te piece of stained glass at te rigt is made up of eigt congruent parallelograms. Eac parallelogram as a ase of 8 centimeters and a eigt of centimeters. Find te area of te entire piece. Error Analysis In Eercises and 7, students were asked to find te area of te parallelogram. Descrie and correct any errors. STAINED GLASS Te amount of glass needed for a stained glass design depends on te overall area of te piece, wile te amount of metal needed to connect te pieces depends on teir individual perimeters.. 7. A (7)() Using Algera In Eercises 8 0, A gives te area of te parallelogram. Find te missing measure. 8. A 88 cm 9. A 0 in. 0. A 9 ft A ()() 77 8 cm ft in.. A parallelogram as a ase of meters and an area of 0 square meters. Find te eigt.. A parallelogram as a eigt of feet and an area of square feet. Find te ase. Area of Romuses Find te area of te romus Area of Parallelograms
6 Puzzles Penrose Romuses Te romuses sown elow are named after te Britis pysicist Roger Penrose. He found tat tese romuses can e used to cover a surface witout any gaps and witout aving to repeat te same pattern. An eample is sown elow PENROSE TILINGS Te jigsaw puzzle aove illustrates a Penrose tiling tat involves si figures. Te design will cover a surface witout gaps and witout repeating te same pattern. 9. Find te area of te yellow romus. 0. Find te area of te red romus.. Te pattern aove contains yellow and red romuses. Find its area. Area of Comple Polygons Find te area of te entire polygon y adding te areas of te smaller polygons.... ft cm ft cm cm 9 ft m m EXAMPLE Find te area of te parallelogram. Area on te Coordinate Plane y Count te squares to find te ase and te eigt. Te ase is units and te eigt is units. A Formula for area of a parallelogram ()() Sustitute for and for. 0 Multiply. Area on te Coordinate Plane Find te area of te parallelogram.. y. y 7. y Capter 8 Polygons and Area
7 Callenge Find te area of te given quadrilateral. 8. Parallelogram ABCD 9. Romus FGHJ A B 8 D 9 C F G 0 0 J H Standardized Test Practice 0. Multiple Coice Wat is te area of te parallelogram? A cm B cm cm C 8 cm D 0 cm cm. Multiple Coice Wat is te area of te romus? F ft G ft ft H 7 ft J 8 ft 7 ft 7 ft ft. Multiple Coice Wat is te area of te saded polygon? A 0 in. B 0 in. C 0 in. D 0 in. in. 8 in. in. Mied Review Similar Triangles Find te value of. (Lesson 7.)... 8 Conve and Concave Polygons Decide weter te polygon is conve or concave. (Lesson 8.) Algera Skills Simplifying Radicals Simplify te epression. (Skills Review, p. 8) Area of Parallelograms
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