SURFACE AREA and VOLUME

SURFACE AREA and VOLUME

Solid Figures Introduction

PRISMS CYLINDERS

PYRAMIDS

CONESSPHERES

APPENDIX


APPENDIX

SOLID FIGURES

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In a previous unit we discussed POLYGONS. POLYGONS are 2-dimensional objects.

That means they have length and width, but no depth.

Like they are cut out of paper.


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SOLID FIGURES

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In this unit, we will learn about SOLIDS. A SOLID is a 3-dimensional (3D) object. That means it has length, width AND depth.


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SOLID FIGURES

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When the SOLID is made up of POLYGONS, it is called a POLYHEDRON

Each POLYHEDRON is made up ofSurfaces (called faces)

Segments (called edges)

Corners (called vertices)


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SOLID FIGURES

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There are many types of solids, we are going to study the 5 basic

types:PRISM PYRAMID

CYLINDER CONE SPHERE

Before we can do

anything with

SOLIDS, we must

understand the

difference between

these shapes


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SOLID FIGURES

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PRISMS: To create a prism, draw any polygon you like. Next, draw another polygon, just like the first one

BASE BASE BASE

Connect the corresponding vertices


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SOLID FIGURES

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PRISMS:


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SOLID FIGURES

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PYRAMIDS: Pyramids have one baseInstead of connecting to an identical base…They connect to a point.

BASE BASE BASE


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SOLID FIGURES

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Pyramids:


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SOLID FIGURES

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When we name prisms and pyramids, we describe the base to tell what kind of prism or pyramid it is.

BASE PRISM PYRAMIDTriangle Triangular Prism Triangular PyramidRectangle Rectangular Prism Rectangular

PyramidPentagon Pentagonal Prism Pentagonal PyramidHexagon Hexagonal Prism Hexagonal PyramidHeptagon Heptangonal Prism Heptangonal

PyramidOctagon Octagonal Prism Octagonal PyramidDecagon Decagonal Prism Decagonal Pyramid


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SOLID FIGURES

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CYLINDERS: A cylinder is like a prism…But the bases are circles


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SOLID FIGURES

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Cylinders:


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SOLID FIGURES

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CONES: A cone is a circular pyramid.


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SOLID FIGURES

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Cones:


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SOLID FIGURES

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SPHERES:

A sphere is a unique shape...It has no base or sides or edges or vertices.


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SOLID FIGURES

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SPHERES:


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SOLID FIGURES

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Identify each of the following shapes as a PRISM, PYRAMID, CYLINDER, CONE or SPHERE #1 #2 #3 #4

#5 #6

#7

#8 #9 #10

SPHERE CYLINDERPRISM

PYRAMID

CYLINDER CONE PYRAMID

PRISM SPHERE CONE & SPHERE

http://www.freakingnews.com/images/adv-images/cone.jpg


APPENDIX

SOLID FIGURES

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So what are we going to do with these shapes?

We are going to calculate



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SOLID FIGURES

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SUFARCE AREA:Is literally the area of all the surfaces of a solid.


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SOLID FIGURES

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VOLUME:Is the measure of how much “space” an object occupies.It is often thought of as how much water an object can hold.


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SOLID FIGURES

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VOLUME:Or how many cubic units it would take to fill an object

This is why volume is measured in CUBIC UNITS

Like:

Cubic feet: ft3

Cubic centimeters cm3


APPENDIX

PRISMS

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A prism is a solid with 2 congruent BASES connected by LATERAL FACES

Lateral face


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PRISMS

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To find the surface area of a PRISM, you can do it 2 ways:Method 1:

Find the area of each shape that makes up the Solid, and add them together

26

10

The front 2 x 10 = 20

20

The back 2 x 10 = 20The left 6 x 10 = 60The right 6 x 10 = 60The top 2 x 6 = 12The bottom 2 x 6 = 12

60

12

= 184

(in a rectangular prism any pair of opposite sides can be bases)


APPENDIX

PRISMS

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To find the surface area of a PRISM, you can do it 2 ways:Method 2: Use the formula for SURFACE AREA:

26

10

BASE

•Identify the BASEs

•Find the Area and Perimeter of the base

•

B: area of the base P:

perimeter of the base

h: height

of the

prism

phBASAreaSurface

2.).(1016122 SA

16024184

12

16

10

P; 16A: 12


APPENDIX

PRISMS

MENU

A: 270m2

6m

Find the surface area of this regular decagonal prism:

4m

phBSA 2

P:60m

4602702 SA

Find the perimeter of the base6+6+6+6+6+6+6+6+6+6

=60

Area of the

basesLateral Area

2405402780m


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PRISMS

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Finding Volume is easier!Find the VOLUME of this regular decagonal prism:

A: 270m2

6m4m

P:60m

hBVolume :

B: area of the base

h: height

of the

prism

4270Volume31080m


APPENDIX

PRISMS

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RIGHT PRISMS vs OBLIQUE PRISMSA right prism is a prism where the lateral faces are all perpendicular to the bases.

An oblique prism is a prism where the lateral faces are NOT perpendicular to the bases.


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PRISMS

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SUMMARY:Things you need to know about a prism:

A prism has 2 bases

All the lateral faces are rectangles

Surface Area = 2B + phB is the area of the basep is the perimeter of the

baseh is the height of the prismp x h gives the lateral area

Volume = B x hB is the area of the baseh is the height of the prism


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PRISMS

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Find the……Surface Area

…Lateral Area

…Volume

2cm

4cm 3cm

A:P:

6cm2

12cm

First we have to find that missing sideUse the Pythagorean theorem

4cm 3cm

5cm


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PRISMS

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4cm 3cmFind the……Surface Area

…Lateral Area

…Volume

5cm

A:P:

6cm2

12cm

hB

phB 221262

2cm

2412236cm

224cm

26312cm


APPENDIX

PRISMS

MENU

Find the surface area and volume.

Area:Perimeter:

814

6

10

11224

136ft2

52ft

phBSA 25521362 SA

260272SA2532 ft

hBV

5136V3680 ft


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PRISMS

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Find the volume.

81011

8149 1008

880

880100821888u


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PRISMS

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Archimedes problem.


APPENDIX

CYLINDERS

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Since CYLINDERS are like prisms with circular bases,

We need to remember a few basic rules for circles.


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CYLINDERS

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Find the Circumference and area of the circle:

12in6in

The red segment is the DIAMETER

rnceCircumfere 2

2rArea

We need the RADIUS

6212

in68.37

26

36204.113 in


APPENDIX

CYLINDERS

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A CYLINDER is like a prism with circular basesThe surface area of a cylinder can be found with this formula

chBSA 2

h: height of

the cylinder

Most people find this formula easier: rhrSA 22 2

r

hArea of

the bases

Lateral Area

B: area of the base

C: circumference

of the base


APPENDIX

CYLINDERS

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Find the SURFACE AREA of this cylinder:

ft3

ft10

rhrSA 22 2

103232 2 6018

78292.244 ft


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CYLINDERS

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The VOLUME of a Cylinder:The VOLUME of a cylinder can be found with this formula

BhV

h: height of

the cylinder

The expanded version of the formula:

2V r h

r

h

B: area of the base


APPENDIX

CYLINDERS

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Find the VOLUME of this cylinder:

ft3

ft10

2V r h23 10

9 10 90

3282.6f t


APPENDIX

CYLINDERS

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SUMMARY

r

h2V r h

What you need to know about a cylinder:•A cylinder is a prism with circular bases

•The lateral area is a rectangle

•Surface Area:

•Volume:

•Lateral Area:

rhrSA 22 2

rhLA 2


APPENDIX

CYLINDERS

MENU

Find the Lateral Area, Surface Area and Volume:

11mm

9mm

2V r hrhrSA 22 2

rhLA 29112

198272.621 mm

9112112 2 198242

440 26.1381 mm

211 9 33419.46mm


APPENDIX

CYLINDERS

MENU

Find the surface area and volume of this shape 31ft

8ft

rhrSA 22 2 If it was a complete cylinder:

318282 2 236.1959 ft

But we only want half of it268.979 ft

What did we forget?

31ft16ft

49668.979 268.1475 ft


APPENDIX

CYLINDERS

MENU

Find the surface area and volume of this shape 31ft

8ft

If it was a complete cylinder: 2V r h

28 31 36,229.76f t

But we only want half of it33,114.88f t


APPENDIX

PYRAMIDS

MENU

A PYRAMID is a shape with a single base, where all the vertices of the base are connected to a single point (apex)


APPENDIX

PYRAMIDS

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Just like with prisms, we will need to find:•Base Perimeter•Base Area•Height of the pyramid

However, with pyramids, we will also need to find a different measurement……SLANT HEIGHThe

ight

Slant height


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PYRAMIDS

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SLANT HEIGHT

Slant height is the distance someone would travel walking up the OUTSIDE of a pyramid.


APPENDIX

PYRAMIDS

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http://www.myspacegens.com/files/animated_gif/download.php?id=3997021673


APPENDIX

PYRAMIDS

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Finding the SURFACE AREA of a PYRAMID

PlBSA21

B: area of the base

P: perimeter of the base

L: Slant

height

Rectangular Pyramid

22m

22m

25m


APPENDIX

PYRAMIDS

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Finding the SURFACE AREA of a PYRAMID

PlBSA21

Rectangular Pyramid

1484 88 252

Base Area:Base Perimeter:

484m2 88m

484 1100 Lateral Area21584m 22

m22m

25m


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PYRAMIDS

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Finding the VOLUME of a PYRAMID

hBV 31

B: area of the base

h: height

Rectangular Pyramid

10ft 8ft

11ft1180

31

V

Base Area:Base Perimeter:

80ft2

36ft3293.3f t


APPENDIX

PYRAMIDS

MENU

SUMMARY:Things you need to know about a pyramid:

A pyramid has only 1 base

All the lateral faces are triangles

Surface Area =B is the area of the basep is the perimeter of the basel is the slant height of the

pyramid½ x p x l is the lateral area

Volume =B is the area of the baseh is the height of the prism

PlBSA21

hBV 31

h

l


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PYRAMIDS

MENU

Hey, I need to know the height. Hey, I need to know the slant height.

24

20

222 2012 h400144 2 h2562 h16h

16

15

222 158 L2289 LL17

Lateral Area


APPENDIX

PYRAMIDS

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Find the Lateral Area, Surface area and Volume:

PlBSA21

PlLA21

B:P:

40080

268021

21040u


APPENDIX

PYRAMIDS

MENU


PlBSA21

B:P:

40080

1400 80 262SA

21440SA u


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PYRAMIDS

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B:P:

40080

13V B h

1 4003V h 10

h

2 2 210 26h 2100 676h 2 576h

24h

1 400 243V

33200V u


APPENDIX

PYRAMIDS

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Find the Volume:

264 48

111231

44

4448292u


APPENDIX

CONES

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A CONE is basically a pyramid with a circular base


APPENDIX

CONES

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R

HL

R: radius

H: height

L: slant height

lrBSA

hrV 2

31

Lateral Area

lrrSA

or

2


APPENDIX

CONES

MENU

13

12i

n

5in

Find the slant height with the Pythagorean Theorem

Find the surface area and volume for the cone shown: 2SA r r l

13525 6525

90 2282.6in

213V r h

12531 2

100 3314in


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CONES

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Find the surface area and volume:

hrV 2

31

lrrSA 2

15992 224.678 u

12931 2

208.3052 u


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CONES

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Find the Lateral Area:

lrBSA

Lateral Area

5

5321.47 u


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CONES

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Find the Volume:

Cylinder:

Cone:

TOTAL:hrV 22

18102 2 311304m

hrV 2

31

121031 2

31256m

3560,12 m


APPENDIX

SPHERES

MENU

A sphere is just a ball.

It has 1 important measurement, that is used to find all its other propertiesThe RADIUS R

R

Official definition of a SPHERE:The collection of all points in space that

are the same distance (radius) from a point (center).


APPENDIX

SPHERES

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12m

3

34 rV

24 rSA 2124 576

264.1808 m

31234

2304

356.234,7 m


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SPHERES

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Find the surface area:

24S r24 (8)

24 64 804 .in 8 in.


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SPHERES

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Find the volume:

32 (3.14)3

343

V r

34 (2)3

4 (8)3

323

334 .ft

2 ft.

10in


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SPHERES

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Find the volume and surface area of the hemisphere:

24 rSA 2104

400

2rA 210

100

100200 3002942in

Half the sphere +the circle

33.2093 in

10in


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SPHERES

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Find the volume and surface area of the hemisphere:

3

34 rV 24 rSA

2942in

31034

37.4186 inWe only want half


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SPHERES

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Find the volume:

Cone:

Ice cream:

TOTAL:

21

34 3

rV

hrV 2

31 125

31 2

3314mm

215

34 3

367.261 mm

67.261314

67.575

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