Lesson 11-1

Areas of Parallelograms

Objectives

• Find perimeters and areas of parallelograms– P = 2 (l + w)

– A = b·h

• Determine whether points on a coordinate plane define a parallelogram

Vocabulary

• base – the “horizontal” distance of the parallelogram (bottom side)

• height – the “vertical” distance of the parallelogram

• area – the amount of flat space defined by the figure (measured in square units)

• perimeter – once around the figure

Area of Parallelograms

A B

C D

h

Parallelogram PerimeterAdd length of all four sidesP = AB + DB + DC + CA (or P = 2AB + 2BD)

Parallelogram AreaA = h · bh is heightb is base (AB or CD)(similar to a rectangle)

A B

C D

b

Example 1:

A B

C D

10

15

12

Find the perimeter and area of parallelogram ABCD

P = 2(12 + 15) = 2(27) = 54

A = bh = 15(10) = 150 square units

Example 2:M N

P O6

15

10

Find the perimeter and area of parallelogram MNOP

9

P = 2(10 + 15) = 2(25) = 50

A = bh = 15(h) = 15h square units

10² = 6² + h²

100 – 36 = 64 = h²

√64 = 8 = h

So, area = 15(8) = 120 square units

Base and Side: Each pair of opposite sides of a parallelogram has the same measure. Each base is 32 inches long, and each side is 24 inches long.

Find the perimeter and area of .

Perimeter: The perimeter of a polygon is the sum of the measures of its sides. So, the perimeter of RSTU is

2(32) + 2(24) = 112 inches.

12 = x Divide each side by 2.

Height: Use a 30-60-90 triangle to find the height. Recall that if the measure of the leg opposite the 30 angle is x, then the length of the hypotenuse is 2x, and the length of the leg opposite the 60 angle is x√3 .

24 = 2x Substitute 24 for the hypotenuse.

So, the height of the parallelogram is x√3 or12 √3 inches.

Area:

Answer: The perimeter of RSTU is 112 inches, and the area is about 665.1 square inches.

Example 3:

D E

G F

27

60°

Find the perimeter and area of parallelogram DEFG

21

P = 2(21 + 27) = 2(48) = 96

A = bh = 27(h) = 27h square units

(side opposite 60°) h = ½ hyp √3

h = ½ (21) √3

h = 10.5 √3So, area = 27(10.5√3) ≈ 491 square units

h

The Kanes are planning to sod some parts of their yard. Find the number of square yards of grass needed.

To find the number of square yards of grass needed, find the number of square yards of the entire lawn and subtract the number of square yards where grass will not be needed. Grass will not be needed for the vegetable garden, the garage, or the house and walkways.

Entire lawn:

Vegetable Garden:

Garage:

House and Walkways:

Area

Entire Lawn

Vegetable Garden Garage

House and Walkways

The total area is 30,000 – 2000 – 3000 – 6000 or 19,000 square feet. There are 9 square feet in one square yard, so divide by 9 to convert from square feet to square yards.

Answer: They will need about 2111 square yards of sod.

The Wagners are planning to put hardwood floors in their dining room, living room, and kitchen. Find the number of square yards of wood needed.

Answer:

Summary & Homework

• Summary:– The area of a parallelogram is the product of the

base and the height

• Homework: – pg 598-600; 9-16, 27, 28