section 8.5. find the area of parallelograms. base of a parallelogram height of a parallelogram...
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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.
base
base
baseba
se
The 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.
b
h
Area of Parallelogram
Words: Area =(base)(height)
Symbols: A = bh
Very Important: The height must be perpendicular to the base.
Find the Area of a ParallelogramExample 1
Find the area of the parallelogram.
SOLUTION
Use the formula for the area of a parallelogram.Substitute 9 for b and 6 for h.
A = bh Formula for the area of a parallelogram
= (9)(6) Substitute 9 for b and 6 for h.
= 54 Multiply.
The parallelogram has an area of 54 square meters.
ANSWER
Find the Height of a ParallelogramExample 2
Find the height of the parallelogramgiven that its area is 78 square feet.
SOLUTION
A = bh Formula for the area of a parallelogram
78 = 12h Substitute 78 for A and 12 for b.
6.5 = h Divide each side by 12.
The parallelogram has a height of 6.5 feet.ANSWER
Your Turn:
In Exercises 4–6, A gives the area of the parallelogram. Find the missing measure.
ANSWER h = 6 in.
ANSWER b = 6 m
ANSWER h = 4 cm
4. A = 72 in.2
5. A = 30 m2
6. A = 28 cm2
Rhombus
A parallelogram with opposite equal acute and obtuse angles and four equal sides.
Diagonals
Diagonals
4 equal sides
Special “Rhombus Rule”Since the diagonals are perpendicularAnother way to find the area of a rhombus is:Area = ½ (product of the diagonals)
d 1
d2
area = ½ (d1 . d2)
This is good when you only know the diagonals, but not the sides or height
Area of a Rhombus
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) = 6Area of 4 triangles = 4(6) = 24Notice that 1/2d1d2 or ½(6)(8), also equals 24
The formula for the area of a rhombus can be justified using the area of a triangle. A specific case follows.