Geometry
11.2 Areas of Parallelograms,
Rhombuses, and Triangles
Parallelogram
Height
Any side of the parallelogram
Base
A = bh
The length of the altitude.
The altitude is defined as any segment perpendicular to the line
containing the base from any point on the opposite side.
Parallelogram
Check this out!
Height
Base
Any side of the parallelogram
A = bh
You would find the same area either
way you solved!
Perpendicular to the base(altitude).
Solve.
1. Find the area of a parallelogram with base 6 cm and corresponding height 7 cm.
A = 6(7)
A = 42 units2
2. Find the area of a parallelogram with base 6√2 and corresponding height 10√2 .
A = (6√2)(10√2)
A = 120 units2
Let’s do #4,5!
You try #6!
Find the area of each parallelogram.
3. Base 12 and height 8.
A = 12(8)
A = 96 units2
5.
4.
6 2
6.
4
10
5√3
12
6
A = 12(4)
A = 48 units2
6
45
60
5
6√2
15
A = 15(5√3)
A = 75√3 units2
A = 6(6√2)
A = 36√2 units2
Triangle
Imagine dropping a rock from the highest
point down to the base to find the height.
Height
Base
A = ½ bh
½ the base times the height
or
½ the height times the base
WHICHEVER IS EASIER!
WHY IS THIS THE FORMULA?
Find the area of each figure.
7.
Pythagorean Theorem/Triples
Let’s do #7!
8.
This is an altitude.
20
7
15
Dropping a rock!
25
4
24
5
Total area = area of top triangle + area of bottom triangle
A = ½ (15)(20) + ½ (24)(7)
A = 5(2)
A = 10 units2
A = (10)(15) + (12)(7)
A = 150 + 84
A = 234 units2
9. A triangle with base 18 and height 7. A = 9(7)
10. A triangle with sides 5, 12, 13.
12
It is a right triangle.
Let’s do #10,12!
You try #13!
A = 63 units2
13
A = ½ (12)(5)
A = 30 units2
5
11. Find the area of an isosceles triangle with sides 30, 30, and 24.
h2 + 122 = 302
Area = 12(6√21)
30
30
2
h
h + 144 = 900
h2 = 756
h = 6√21
12
Area = 72√21 units2
24
12. Find the area of an isosceles triangle with base 16 and perimeter 52.
h2 + 82 = 182
Area = 8(2√65)
18
18
2
h
h + 64 = 324
h2 = 260
8
Area = 16√65 units2
16
h = 2√65
13. Find the area of an equilateral triangle with sides 12 cm.
12
6√3
60o
6
12
Area = 6(6√3)
Area = 36√3 units2
Area = 6(6√3)
6√3
o
60
12
6
14. Find the area of an equilateral triangle with height 6√3 .
Area = 36√3 units2
Rhombus
A = ½ d1d2
Take ½ of whichever diagonal is easier than multiply.
Let’s do #16,18!
You try #17!
Find the area of each rhombus.
15.
16.
10
12
A rhombus is a parallelogram.
8
15
15
45o
4 3
4 3
A = ½ d1d2
A = 30(8)
A = 240 units2
A = ½ d1d2
A = 10(24)
A = 240 units2
18.
17
12
10
17.
60
10√2
135
10
4
10 2
A = ½ d1d2
A = bh
A = 4(8√3)
A = 10(10√2)
2
A = 32√3 units A = 100√2 units2
Let’s do #20!
You try #21!
19. Find the area of a rhombus with diagonals 8 m and 20 m.
A = 8(10) = 80 units2
20. Find the area of a rhombus with perimeter 52 and one diagonal 10.
13
13
12
5
A = 12(10) = 120 units2
5
13
13
21. Find the area of a rhombus with perimeter 100 and one diagonal 14.
25
25
24
7
25
A = 14(24) = 336 units2
7
25
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.
A = short height(long base)
A = short base(long height)
The longer altitude is 10 units.
The area is 14(5) = 70 units2.
Since A = bh
70 = 7h
10 = h
HW
• P. 431 WE (1-21 odd)
P. 426-427 (20-30 Even)
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