Geometry

11.2 Areas of Parallelograms,

Rhombuses, and Triangles

Parallelogram

A = bh

The length of the altitude.

Base

Height

Any side of the parallelogram

The altitude is defined as any segment perpendicular to the linecontaining the base from any point on the opposite side.

Parallelogram

A = bh

Perpendicular to the base(altitude).

Base

Height

Any side of the parallelogram

Check this out!

You would find the same area eitherway you solved!

Solve.1. Find the area of a parallelogram with base 6 cm and corresponding height 7 cm.

2. Find the area of a parallelogram with base 6√2 and corresponding height 10√2 .

A = 6(7)

A = 42 units2

A = (6√2)(10√2)

A = 120 units2

Find the area of each parallelogram.

3. Base 12 and height 8.

4

6

12

15

10

4560

6 24. 5. 6.

A = 12(8)A = 96 units2

A = 12(4)A = 48 units2

5

5√3

A = 15(5√3)A = 75√3 units2

66√2

A = 6(6√2)A = 36√2 units2

Let’s do #4,5!You try #6!

Triangle

A = ½ bh

Base

Height

½ the base times the height

or½ the height times the base

WHICHEVER IS EASIER!

Imagine dropping a rock from the highestpoint down to the base to find the height.

WHY IS THIS THE FORMULA?

Find the area of each figure.

20

25

7

4

5

7. 8.Pythagorean Theorem/Triples

2415

Total area = area of top triangle + area of bottom triangle

A = ½ (15)(20) + ½ (24)(7)

A = (10)(15) + (12)(7)

A = 150 + 84

A = 234 units2

This is an altitude.

Dropping a rock!

A = 5(2)

A = 10 units2

Let’s do #7!

9. A triangle with base 18 and height 7.

10. A triangle with sides 5, 12, 13.

11. Find the area of an isosceles triangle with sides 30, 30, and 24.

12. Find the area of an isosceles triangle with base 16 and perimeter 52.

13. Find the area of an equilateral triangle with sides 12 cm.

14. Find the area of an equilateral triangle with height 6√3 .

A = 9(7) A = 63 units2

It is a right triangle.

5

12 13

A = ½ (12)(5) A = 30 units2

30 30

2412

hh2 + 122 = 302

h2 + 144 = 900

h2 = 756

h = 6√21

Area = 12(6√21)

Area = 72√21 units2

18 18

168

hh2 + 82 = 182

h2 + 64 = 324

h2 = 260

h = 2√65

Area = 8(2√65)

Area = 16√65 units2

60o

12 12

126

6√3Area = 6(6√3)

Area = 36√3 units2

60o

6

6√3Area = 6(6√3)

Area = 36√3 units2

Let’s do #10,12!You try #13!

Rhombus

A = ½ d1d2

Take ½ of whichever diagonal is easier than multiply.

Find the area of each rhombus.

10

12

817

15. 16. 17. 18.

60

4 3

4 3135

10 2

A = ½ d1d2

10

12

A = 10(24)A = 240 units2

1515

A = ½ d1d2

A = 30(8)A = 240 units2

4

A = ½ d1d2

A = 4(8√3)A = 32√3 units2

A rhombus is a parallelogram.

45o

10√2 10

A = bhA = 10(10√2)

A = 100√2 units2

Let’s do #16,18!You try #17!

19. Find the area of a rhombus with diagonals 8 m and 20 m.

20. Find the area of a rhombus with perimeter 52 and one diagonal 10.

21. Find the area of a rhombus with perimeter 100 and one diagonal 14.

A = 8(10) = 80 units2

13

1313

13

55

12

A = 12(10) = 120 units2

25

2525

25

77

24

A = 14(24) = 336 units2

Let’s do #20!You try #21!

Bonus

• A parallelogram has two bases and two altitudes. Its longer base is 14 and its shorter altitude is 5. If its shorter base is 7, find its longer altitude.

The area is 14(5) = 70 units2.

Since A = bh

70 = 7h

10 = h

The longer altitude is 10 units.

A = short base(long height)A = short height(long base)

HW

• P. 431 WE (1-21 odd)

P. 426-427 (20-30 Even)

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