Area of Parallelograms

Section 8.5

Goal

Find the area of parallelograms.

Key Vocabulary

Base of a parallelogramHeight of a parallelogramParallelogramRhombus

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.

What is the area of this parallelogram?

b=base

h=height

Area of Parallelogram

What is the area of this parallelogram?

CUT HERE!

b=base

h=height

Area of Parallelogram

What is the area of this parallelogram?

MOVE TO HERE!

b=base

h=height

Area of Parallelogram

It’s the same as the area of this rectangle

h=height

b=base

Area of Parallelogram

Arearectangle= base x height(perpendicular height h)

h=height

b=base

Area of Parallelogram

Areaparallelogram = base x height(perpendicular height h)

b=base

h=height

Area of Parallelogram

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:

Find the area of the parallelogram.

3.

2.

1.

ANSWER 96 yd2

ANSWER 77 mm2

ANSWER 196 ft2

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

Area of a Rhombus

Same as Area of parallelogram

A = b . h

b

h

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.

Find the Area of a RhombusExample 3

Find the area of the rhombus.

a. b.

= 108

The area of the rhombus is 70 square inches.

The area of the rhombus is 108 square inches.

SOLUTION

A = d1d221a.

SOLUTION

b. A = d1d221

21 (14)(10)= (6 + 6)(9 + 9)= 2

1

= 70 (12)(18)= 21