Area of Parallelograms

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Area of Parallelograms. Section 8.5. Goal. Find the area of parallelograms. Key Vocabulary. Base of a parallelogram Height of a parallelogram Parallelogram Rhombus. Parallelogram. A parallelogram is a quadrilateral where the opposite sides are congruent and parallel.
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Area of ParallelogramsSection 8.5Goal
  • Find the area of parallelograms.
  • Key Vocabulary
  • Base of a parallelogram
  • Height of a parallelogram
  • Parallelogram
  • Rhombus
  • Parallelogram
  • A parallelogram is a quadrilateral where the opposite sides are congruent and parallel.
  • A rectangle is a type of parallelogram, but we often see parallelograms that are not rectangles (parallelograms without right angles).
  • The Base of a Parallelogram
  • Either pair of parallel sides of a parallelogram are called the bases of the parallelogram.
  • basebasebasebaseThe Height of a Parallelogram
  • The shortest distance (perpendicular distance) between the bases of a parallelogram is called the height of the parallelogram.
  • The height of the parallelogram is always perpendicular to the bases.
  • Area of ParallelogramWhat is the area of this parallelogram?h=heightb=baseArea of ParallelogramWhat is the area of this parallelogram?h=heightb=baseCUT HERE!Area of ParallelogramWhat is the area of this parallelogram?h=heightb=baseMOVE TO HERE!Area of ParallelogramIt’s the same as the area of this rectangleh=heightb=baseArea of ParallelogramArearectangle= base x height(perpendicular height h)h=heightb=baseArea of ParallelogramAreaparallelogram = base x height(perpendicular height h)h=heightb=baseArea of Parallelogram
  • Words: Area =(base)(height)
  • Symbols: A = bh
  • hbVery Important: The height must be perpendicular to the base.SOLUTIONUse the formula for the area of a parallelogram.Substitute 9 for band 6 for h.A = bhFormula for the area of a parallelogram=(9)(6)Substitute 9 for b and 6 for h.=54Multiply.The parallelogram has an area of 54square meters.ANSWERExample 1Find the Area of a ParallelogramFind the area of the parallelogram.A =bhFormula for the area of a parallelogram78=12hSubstitute 78 for A and 12 for b.6.5 =hDivide each side by 12.The parallelogram has a height of 6.5 feet.ANSWERExample 2Find the Height of a ParallelogramFind the height of the parallelogramgiven that its area is 78 square feet.SOLUTIONYour Turn:1.96 yd2ANSWER2.77 mm2ANSWER3.196 ft2ANSWERFind the area of the parallelogram.Your Turn:4.A = 72 in.2h =6 in.ANSWERA = 30 m25.b =6 mANSWER6.A = 28 cm2h =4 cmANSWERIn Exercises 4–6, Agives the area of the parallelogram. Find the missing measure.Rhombus
  • A parallelogram with opposite equal acute and obtuse angles and four equal sides.
  • Diagonals 4 equal sidesDiagonals hbArea of a RhombusSame as Area of parallelogramA = b . hd2d1Special “Rhombus Rule”
  • Since the diagonals are perpendicular
  • Another way to find the area of a rhombus is:
  • Area = ½ (product of the diagonals)
  • area = ½ (d1.d2)This is good when you only know the diagonals, but not the sides or heightArea of a RhombusThe formula for the area of a rhombus can be justified using the area of a triangle. A specific case follows.
  • The diagonals divide a rhombus into 4 congruent right triangles. So, the area of the rhombus is 4 times the area of one of the right triangles.
  • Area of 1 triangle = 1/2bh = ½(3)(4) = 6
  • Area of 4 triangles = 4(6) = 24
  • Notice that 1/2d1d2 or ½(6)(8), also equals 24
  • Find the area of the rhombus.b.a.1111122222Example 3Find the Area of a RhombusSOLUTIONSOLUTIONa.b.A=d1d2A=d1d2= (6 + 6)(9 + 9)=(14)(10)= 70=(12)(18)=108The area of the rhombus is 70 square inches.The area of the rhombus is 108 square inches.
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