SURFACE AREA and VOLUME

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SURFACE AREA and VOLUME. Solid Figures Introduction. CYLINDERS. PRISMS. PYRAMIDS. SPHERES. CONES. APPENDIX. SURFACE AREA and VOLUME. SOLID FIGURES. In a previous unit we discussed POLYGONS. . POLYGONS are 2-dimensional objects. . That means they have length and width, but no depth.

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SURFACE AREA and VOLUME

Solid Figures Introduction

CYLINDERS

PRISMS

PYRAMIDS

SPHERES

CONES

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

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.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

In this unit, we will learn about SOLIDS.

A SOLID is a 3-dimensional (3D) object.

That means it has length, width AND depth.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

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

Each POLYHEDRON is made up of

Surfaces (called faces)

Segments (called edges)

Corners (called vertices)

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

Before we can do anything with SOLIDS, we must understand the difference between these shapes

There are many types of solids,

we are going to study the 5 basic types:

PRISM

PYRAMID

CYLINDER

CONE

SPHERE

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

To create a prism, draw any polygon you like.

PRISMS:

Next, draw another polygon, just like the first one

Connect the corresponding vertices

BASE

BASE

BASE

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

PRISMS:

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

PYRAMIDS:

Pyramids have one base

Instead of connecting to an identical base…

They connect to a point.

BASE

BASE

BASE

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

Pyramids:

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

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

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

CYLINDERS:

A cylinder is like a prism…

But the bases are circles

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

Cylinders:

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

CONES:

A cone is a circular pyramid.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

Cones:

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SPHERES:

A sphere is a unique shape...

It has no base or sides or edges or vertices.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SPHERES:

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

Identify each of the following shapes as a

PRISM, PYRAMID, CYLINDER, CONE or SPHERE

#1

#2

#3

#4

SPHERE

CYLINDER

PRISM

#5

#6

PYRAMID

#7

CYLINDER

CONE

PYRAMID

#9

#10

#8

SPHERE

CONE & SPHERE

PRISM

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

So what are we going to do with these shapes?

We are going to calculate

SURFACE AREA and VOLUME

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

VOLUME:

Is the measure of how much “space” an object occupies.

It is often thought of as how much water an object can hold.

APPENDIX

SURFACE AREA and VOLUME

SOLID FIGURES

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

SURFACE AREA and VOLUME

PRISMS

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

Lateral

face

APPENDIX

BASE

SURFACE AREA and VOLUME

PRISMS

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

The front

2 x 10

= 20

12

The back

2 x 10

= 20

The left

6 x 10

= 60

10

60

The right

6 x 10

= 60

20

The top

2 x 6

= 12

The bottom

2 x 6

= 12

6

= 184

2

APPENDIX

SURFACE AREA and VOLUME

PRISMS

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

Method 2:

Use the formula for SURFACE AREA:

• Identify the BASEs

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

• Find the Area and Perimeter of the base

BASE

P; 16

A: 12

10

h: height of the prism

B: area of the base

10

6

P: perimeter of the base

2

12

16

APPENDIX

SURFACE AREA and VOLUME

PRISMS

Find the surface area of this regular decagonal prism:

Find the perimeter of the base

6+6+6+6+6+6+6+6+6+6

=60

Area of the bases

Lateral Area

A: 270m2

P:60m

6m

4m

APPENDIX

SURFACE AREA and VOLUME

PRISMS

Finding Volume is easier!

Find the VOLUME of this regular decagonal prism:

h: height of the prism

B: area of the base

A: 270m2

P:60m

6m

4m

APPENDIX

SURFACE AREA and VOLUME

PRISMS

RIGHT PRISMS vs OBLIQUE PRISMS

A 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.

APPENDIX

SURFACE AREA and VOLUME

PRISMS

SUMMARY:

Things you need to know about a prism:

A prism has 2 bases

All the lateral faces are rectangles

Surface Area = 2B + ph

B is the area of the base

p is the perimeter of the base

h is the height of the prism

p x h gives the lateral area

Volume = B x h

B is the area of the base

h is the height of the prism

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SURFACE AREA and VOLUME

PRISMS

Find the…

3cm

…Surface Area

4cm

2cm

…Lateral Area

First we have to find that missing side

3cm

4cm

…Volume

Use the Pythagorean theorem

5cm

A:

P:

6cm2

12cm

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SURFACE AREA and VOLUME

PRISMS

Find the…

…Surface Area

3cm

4cm

5cm

2cm

…Lateral Area

…Volume

A:

P:

6cm2

12cm

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SURFACE AREA and VOLUME

PRISMS

Find the surface area and volume.

10

8

112

24

14

6

Area:

Perimeter:

136ft2

52ft

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SURFACE AREA and VOLUME

PRISMS

Find the volume.

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PRISMS

Archimedes problem.

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Since CYLINDERS are like prisms with circular bases,

We need to remember a few basic rules for circles.

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the Circumference and area of the circle:

6in

12in

The red segment is the DIAMETER

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

A CYLINDER is like a prism with circular bases

The surface area of a cylinder can be found with this formula

h: height of the cylinder

B: area of the base

Area of the bases

Lateral Area

C: circumference of the base

Most people find this formula easier:

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the SURFACE AREA of this cylinder:

APPENDIX

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CYLINDERS

The VOLUME of a Cylinder:

The VOLUME of a cylinder can be found with this formula

h: height of the cylinder

B: area of the base

The expanded version of the formula:

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the VOLUME of this cylinder:

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

SUMMARY

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:

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the Lateral Area, Surface Area and Volume:

11mm

9mm

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the surface area and volume of this shape

31ft

8ft

16ft

31ft

If it was a complete cylinder:

What did we forget?

But we only want half of it

APPENDIX

SURFACE AREA and VOLUME

CYLINDERS

Find the surface area and volume of this shape

31ft

8ft

If it was a complete cylinder:

But we only want half of it

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

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

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

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…

height

Slant height

…SLANT HEIGHT

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

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

SLANT HEIGHT

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PYRAMIDS

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PYRAMIDS

Finding the SURFACE AREA of a PYRAMID

Rectangular Pyramid

L: Slant height

B: area of the base

P: perimeter of the base

25m

22m

22m

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Finding the SURFACE AREA of a PYRAMID

Rectangular Pyramid

25m

Lateral Area

22m

22m

Base Area:

Base Perimeter:

484m2

88m

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Finding the VOLUME of a PYRAMID

Rectangular Pyramid

h: height

B: area of the base

11ft

8ft

10ft

Base Area:

Base Perimeter:

80ft2

36ft

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

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 base

p is the perimeter of the base

l is the slant height of the pyramid

½ x p x l is the lateral area

Volume =

B is the area of the base

h is the height of the prism

h

l

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Hey, I need to know the height.

Hey, I need to know the slant height.

15

20

24

16

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Find the Lateral Area, Surface area and Volume:

Lateral Area

B:

P:

400

80

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Find the Lateral Area, Surface area and Volume:

B:

P:

400

80

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Find the Lateral Area, Surface area and Volume:

h

10

B:

P:

400

80

APPENDIX

SURFACE AREA and VOLUME

PYRAMIDS

Find the Volume:

APPENDIX

SURFACE AREA and VOLUME

CONES

A CONE is basically a pyramid with a circular base

APPENDIX

SURFACE AREA and VOLUME

CONES

H: height

L: slant height

Lateral Area

L

H

R

APPENDIX

SURFACE AREA and VOLUME

CONES

Find the surface area and volume for the cone shown:

Find the slant height with the Pythagorean Theorem

13

12in

5in

APPENDIX

SURFACE AREA and VOLUME

CONES

Find the surface area and volume:

APPENDIX

SURFACE AREA and VOLUME

CONES

Find the Lateral Area:

Lateral Area

5

APPENDIX

SURFACE AREA and VOLUME

CONES

Find the Volume:

TOTAL:

Cylinder:

Cone:

APPENDIX

SURFACE AREA and VOLUME

SPHERES

A sphere is just a ball.

It has 1 important measurement, that is used to find all its other properties

R

Official definition of a SPHERE:

The collection of all points in space that are the same distance (radius) from a point (center).

APPENDIX

SURFACE AREA and VOLUME

SPHERES

12m

APPENDIX

SURFACE AREA and VOLUME

SPHERES

Find the surface area:

8 in.

APPENDIX

SURFACE AREA and VOLUME

SPHERES

Find the volume:

2 ft.

APPENDIX

SURFACE AREA and VOLUME

SPHERES

Find the volume and surface area of the hemisphere:

10in

Half the sphere

+

the circle

APPENDIX

SURFACE AREA and VOLUME

SPHERES

Find the volume and surface area of the hemisphere:

10in

We only want half

APPENDIX

SURFACE AREA and VOLUME

SPHERES

Find the volume:

Cone:

Ice cream:

TOTAL:

APPENDIX

SURFACE AREA and VOLUME

APPENDIX

OPENERS

REVIEWS

ASSIGNMENTS

SURFACE AREA and VOLUME

OPENERS

ABCDEFGHIJKLMNOPQRSTU

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OPENERS

A

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B

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C

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D

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E

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F

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G

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h

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I

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J

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K

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L

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M

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N

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O

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P

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Q

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R

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OPENERS

S

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OPENERS

T

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REVIEWs

APPENDIX