1

Standards 8, 10, 11

Classifying Solids

PROBLEM 1 PROBLEM 2

PROBLEM 3

Parts of a Prism

Classifying Prisms

Surface Area of Prisms

Volume of a Right Prism

Reviewing Perimeters

PROBLEM 4

PROBLEM 5 PROBLEM 6

PROBLEM 7

PRESENTATION CREATED BY SIMON PEREZ RHS. All rights reservedEND SHOW

So,What are Cubic Units

Algebra Integration Activity

2

Standard 8:

Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.

Estándar 8:

Los estudiantes saben, derivan, y resuelven problemas involucrando perímetros, circunferencia, área, volumen, área lateral, y superficie de área de figuras geométricas comunes.

Standard 10:

Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

Estándar 10:

Los estudiantes calculan áreas de polígonos incluyendo rectángulos, triángulos escalenos, triángulos equiláteros, rombos, paralelogramos, y trapezoides.

Standard 11:

Students determine how changes in dimensions affect the perimeter, area, and volume of common geomegtric figures and solids.

Estándar 11:

Los estudiantes determinan cambios en dimensiones que afectan perímetro, área, y volumen de figuras geométricas comunes y sólidos.

3

PRISM PYRAMID

CYLINDER SPHERECONE

Standards 8, 10, 11SOLIDS

4

VERTICES:

A

A,B

B,C

C,D

D,

E

E,

F

F,

G

G,

H

H,

I

I,

J

J.

FACES:

Quadrilaterals HGBC, GFAB, FJEA,

JIDE, IHCD.

Pentagons EDCBA, JIHGF.

EDGES:

CB, BA, AE, ED, DC,

CH, BG, AF, EJ, DI,

HG, GF, FJ, JI, IH.

Standards 8, 10, 11

5

VERTICES:

A

A,B

B,C

C,D

D,

E

E,

F

F,

G

G,

H

H,

I

I,

J

J.

FACES:

Quadrilaterals HGBC, GFAB, FJEA,

JIDE, IHCD.

Pentagons EDCBA, JIHGF.

EDGES:

CB, BA, AE, ED, DC,

CH, BG, AF, EJ, DI,

HG, GF, FJ, JI, IH.

Standards 8, 10, 11

6

PRISM PRISM

PRISM PRISM

CUBE

PRISM

TRIANGULAR RECTANGULAR PENTAGONAL

HEXAGONAL OCTAGONAL

Standards 8, 10, 11

7

SURFACE AREA OF PRISMS

ab

c

cd

d

e

e ff

h

a b

af

e

h

base

base

L = ah + bh + ch + dh + eh + fh

P=

a

a

b

+ b

c

+ c

d

+ d

e

+ e

f

+ f

Perimeter of base:

Lateral Area:

L= ( )ha + b + c + d + e + f

Substituting

L= Ph

Total Area= Lateral Area + 2 Base Area

T = Ph + 2BWhere:

Standards 8, 10, 11

P=perimeterh=height

B= base area

T = L + 2B

L= lateral area

8

VOLUME OF A RIGHT PRISM:

h

h

V =

where:

B= Area of the base

h= height

Standards 8, 10, 11

B

B

Bh

9

REVIEWING PERIMETERS

P =

L

L

W

+ W

L

+ L

W

+ W

P = 2L + 2W

P = L + L + W + WP =

X

X +X

X

+X

X

X

+X

P = 8X

X

+X+X

X

X

P = 6X

X

P = 5X

X

P =

Y

Y

Y

+Y

X

+X

P= 2Y + X

P =4X

X

+X

X

+X

Standards 8, 10, 11

10

Find the lateral area, the surface area and volume of a right prism with a height of 20in and a perimeter of 18 in. whose base is an equilateral triangle.

20 inh=

P= 18 in

Lateral Area:

L= Ph

L= ( )( )18 in 20 in

L= 360 in2

Base Area:

30°

60°

P= 3X

X

18=3X3 3X= 6

63

33

then the base area:B = ( 6)1

2 3 3

= 3 3 3

= 9 3

Total Area= Lateral Area + 2 Base Area

T = L + 2B

15.6 inB 2

T = 360 in + 2(15.6 in )2 2

T = 360 in + 31.2 in2 2

T = 391.2 in2

Surface Area:

Volume:

V=Bh

V= ( )( )15.6 in2 20 in

V= 312 in3

Standards 8, 10, 11

11

Find the lateral area and the surface area of a regular pentagonal prism whose height is 18 ft and whose perimeter is 30 ft.

18 fth=

P= 30 ftLateral Area:

L= Ph

L= ( )( )30 ft 18 ft

L= 540 ft2

Base Area: B = Pa12

12= ( )( )30 4

B 60 feet 2

66

a

3

360°5

= 72° 72°2

=36°

Tan 36°= 3a

aTan36°= 3

(a)(a)

Tan36° Tan36°

a= 4

a

3

36°

P= 5X

30=5X5 5X= 6

46

Total Area= Lateral Area + 2 Base Area

T = L + 2B

T = 540 ft + 2( 60 ft )2 2

T = 540 ft + 120 ft2 2

T = 660 ft 2

Surface Area:

Standards 8, 10, 11

12

Standards 8, 10, 11

Find the lateral area, the surface area and the volume of a right prism with a hexagonal base. The side of the base is 15 cm and the height of the prism is 9 cm.

9 cmh=

15cm

P = 6X

Perimeter:

P = 6( ) 15 cm

P = 90 cm

Lateral Area:

L= Ph

L= ( )( )90 cm 9 cm

L= 810 cm2

15

a

15

30°

60°

7.5 3

7.5

30°

60°

7.5

a =7.5 3 a 13

1513

B= Pa12

12= ( )( )90 13

B 585 cm 2

360°6

= 60°

Base Area:

T = L + 2B

T = 810 cm + 2( 585cm )2 2

T = 810 cm + 1170 cm2 2

T =1980 cm2

Surface Area:

Total Area= Lateral Area + 2 Base Area

Volume:

V=Bh

V= ( )( )585 cm2 9 cm

V = 5265 cm 3

13

STANDARDS

11

1

2

2

23

3

3

4

4

4

1x1x1 = 13

= 1

1 CUBED

2x2x2 = 23

= 8

2 CUBED3x3x3 = 3

3= 27

3 CUBED

4x4x4 = 43

=64

4 CUBED

What is the volume for these cubes?

14

Standards 8, 10, 11

Find the lateral surface of a cube whose volume is 2197 cubic feet.

L

L

LB

V = Bh

B = (L)(L)

B= L2

V = L L2

V= L3

2197 feet = L33

2197 feet = L333 3

L=13 feet

Perimeter of the base:

P = 4( )13 feet

P = 52 feet

Lateral Area:

L= Ph

L= ( )( )52 ft 13 ft

L= 676 ft2

13 feet

Note: Lateral Area = Lateral Surface

15

xx

x

x

x 11

1

1 11

x

V =(x)(x)(x) V =(x)(x)(1)

V =(x)(1)(1)

V =(1)(1)(1)

= x 3 = x 2

= x

= 1

Lets find the volume for this prisms:

Can we use this knowledge to multiply polynomials?

16

Multiply:

(x+3)(x+2)(x+1)

(x+2)

(x+1)

(x+3)

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2

17

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2

18

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2

(x+2)

(x+1)

(x+3)

So, a third degree polynomial may be represented GEOMETRICALLY, by the VOLUME OF A RECTANGULAR PRISM, in this case with SIDES (x+3), (x+2) and (x+1).

19

Multiply:

(2x+1)(x+3)(x+4)

(x+3)

(x+4)

(2x+1)

(2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

2x 3

+12+31x2x 3+15x 2

2x +7x + 3

2

20

Multiply:

(2x+1)(x+3)(x+4) (2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=

21

Multiply:

(2x+1)(x+3)(x+4) (2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=

(x+3)

(x+4)

(2x+1)

22

Standards 8, 10, 11SIMILARITY IN SOLIDS

6

4

8

33 2.25

86

= =43

32.25

= 1.3

Both solids are SIMILAR

23

Standards 8, 10, 11SIMILARITY IN SOLIDS

6

5

10

34 2

106 =

53

42

These solids are NOT SIMILAR

=

1.6 = 1.6 = 2

24

Standards 8, 10, 11

The ratio of the perimeter of two similar prisms is 1:4. If the lateral area of the larger prism is 120 mm, what is the lateral area of the smaller prism?

Lateral Area:

L= Ph L = P h 1 1 1 L = P h 2 2 2

PRISM 1 PRISM 2IF THEN

L P h 1 1 1

L P h 2 2 2

=

P1

P2

=h1

h2

= 14

THENL1

=1 14 4120

L1

120=

1

16(120) (120)

=12016

= 7.5 mm2L1

PRISM 1 < PRISM 2

AND IF

They are similar

What can you conclude from the ratio of the areas and the ratio of the perimeters?