Geometry

Unit VIAreas of Triangles, Parallelograms, Trapezoids, Rhombuses and Kites

Area of a Parallelogram

Draw and BE CF perpendicular to AD�������������� �

. Note that ABE DCF . If these two triangles are congruent, then their areas are equal. Consider cutting off ABE and placing it on top of DCF . You can see that the area of parallelogram ABCD is equal to the area of rectangle EBCF. Theorem 6.21: For a parallelogram with base b and height h, the area is given by the formula __b*h___ Note that the height (also called the altitude) is the length of the segment ___perpendicular to the base

from a point on the opposite side_____________________.

A

B C

D E F

Example: Find the area of parallelogram ABCD. Example: Find the value of x. Quad. EFGH is a parallelogram.

A

8

15 D C

B

60°

E

6 8

10

x

H G

F

30

2𝑠=8𝑠=4

𝑠√3=4 √3

𝐴= h𝑏𝐴=15 (4√3)𝐴=60√3𝑢𝑛𝑖𝑡 𝑠2

𝑎𝑟𝑒𝑎𝑜𝑓 𝐸𝐹𝐺𝐻=6∗10𝑎𝑟𝑒𝑎𝑜𝑓 𝐸𝐹𝐺𝐻=6060=8∗ 𝑥𝑥=7.5

Area of a Triangle Any triangle is half of a parallelogram. Theorem 6.22: For a triangle with base b and height h, the area is given by the formula

h

Example: Find the area of ABC to the 1000th. Example: Find the area of an isosceles right triangle that has a hypotenuse of length 20 cm.

10

C

A

B 25

10 y

x

cos 25=𝑥

10

10∗ cos25=𝑥9 .063=𝑥

sin 25=𝑦

10

4 .226=𝑦10∗ sin 25=𝑦

𝐴=12

h𝑏

𝐴=12∗9.063∗4.226

𝐴=19.150𝑢𝑛𝑖𝑡 𝑠2

20

20=𝑎√2𝑎=10 √2

𝑎=10 √2

𝐴=12

h𝑏

𝐴=12∗10√2∗10√2

𝐴=100𝑐𝑚2

Example: Find the area of DEF with vertices D(–1 , –6), E(-1, 3) and F(2, 0).

𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=12

h𝑏9

𝑏𝑎𝑠𝑒=9h h𝑒𝑖𝑔 𝑡=3

𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=12∗9∗3

𝐴𝑅𝐸𝐴𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒=13.5𝑢𝑛𝑖𝑡 𝑠2

Area of a Trapezoid

Draw diagonal XZ . WXZA __________ XYZA __________

So Trap WXYZA __________+__________ = ____________________

Theorem 6.23: For a trapezoid with bases b1 and b2 and height h, the area of a trapezoid is given by the

formula ___________________

Y X

W Z

h

b2

b1

Example: A trapezoid has an area of 108.8 in2 and bases of lengths 12in. and 20in. Find the height of the trapezoid.

𝐴=12

h(𝑏1+𝑏2)

108.8=12

h(12+20)

108.8=12

h(32)

217.6=32 h

6.8=h

12

20

h

Area of Rhombuses and Kites Recall that the diagonals of both rhombuses and kites are __perpendicular_______. Area ABD = ____________ Area BCD = ______________ Theorems 6.24 & 6.25: For a rhombus or kite with diagonals d1 and d2, the

area is given by the formula

D

A

B

C

Example 1: BD= 12 and AC= 18. Find the area of the kite. . Example 2: BD=13.5 and AC= 21. Find the area of the kite.

D

A

B

C

D

A

B

C

𝐴=12(𝑑1)(𝑑2)

𝐴=12(𝑑1)(𝑑2)

12

18𝐴=

12(12)(18)

𝐴=108

13.5

21

𝐴=12(13.5)(21)

𝐴=141.75

Area Formulas

• Area of a triangle:• Area of parallelogram:• Area of a kite:• Area of a rhombus:• Area of a rectangle:– Perimeter of a rectangle:

• Area of a square:– Perimeter of a square:

𝐴=12

h𝑏

𝐴= h𝑏

𝐴=12𝑑1𝑑2

𝐴=12𝑑1𝑑2

𝐴=𝑙∗𝑤𝑃=2𝑙+2𝑤

𝐴=𝑠2

𝑃=4 𝑠