K.V K.V Faridkot Faridkot

9th-B

Harkamalpreet Singh Brar

Topic

Objectives At the end of the lesson the students

should be able;

To find the surface area of a cylinder ..

What is a cylinder?

The term Cylinder refers to a right circular cylinder. Like a right prism, its altitude is perpendicular to the bases and has an endpoint in each base.

PRESENTATION

base

altitude

radius

base

What will happen if we removed the end of the

cylinder and unrolled the body?

Lets find out !!!!

This will happen if we unrolled and removed the end of a

cylinder….

Circumference of the base

h

2Πr2

Notice that we had formed 2 circles and a 1 rectangle….

The 2 circles serves as our bases of our cylinder and the rectangular region represent the body

How can we solved the surface area of a Cylinder?

To solve the surface area of a cylinder, add the areas of the circular bases and the area of the rectangular region which is the body of the cylinder.

This is the formula in order to solved the surface are of a

cylinder.

SA= area of 2 circular bases + are of a rectangle

oR

We derived at this formula..!!

SA=2Πr2 +2Πr

Or

SA=2Πr (r + h)

Find the surface area of a cylindrical water tank given the height of 20m and the radius of

5m? (Use π as 3.14)

Given:

h=20m

r=5m

SA=2πr2 +2πrh

=2(3.14)(5m)2 + 2[(3.14)(5m)(20m)

=157m2 + 628m

SA =785m2

2-Surface Area of a PrismCubes and Cuboids

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The top and the bottom of the cuboid have the same area.

Surface area of a cuboid

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The front and the back of the cuboid have the same area.

Surface area of a cuboid

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The left hand side and the right hand side of the cuboid have the same area.

Surface area of a cuboid

To find the surface area of a shape, we calculate the total area of all of the faces.

Can you work out the surface area of this cubiod?

Surface area of a cuboid

7 cm

8 cm5 cm

The area of the top = 8 × 5

= 40 cm2

The area of the front = 7 × 5

= 35 cm2

The area of the side = 7 × 8

= 56 cm2

To find the surface area of a shape, we calculate the total area of all of the faces.

So the total surface area =

Surface area of a cuboid

7 cm

8 cm5 cm

2 × 40 cm2

+ 2 × 35 cm2

+ 2 × 56 cm2

Top and bottom

Front and back

Left and right side

= 80 + 70 + 112 = 262 cm2

We can find the formula for the surface area of a cuboid as follows.

Surface area of a cuboid =

Formula for the surface area of a cuboid

h

lw

2 × lw Top and bottom

+ 2 × hw Front and back

+ 2 × lh Left and right side

= 2lw + 2hw + 2lh

How can we find the surface area of a cube of length x?

Surface area of a cube

x

All six faces of a cube have the same area.

The area of each face is x × x = x2

Therefore,

Surface area of a cube = 6x2

This cuboid is made from alternate purple and green centimetre cubes.

Checkered cuboid problem

What is its surface area?

Surface area

= 2 × 3 × 4 + 2 × 3 × 5 + 2 × 4 × 5

= 24 + 30 + 40

= 94 cm2

How much of the surface area is green?

48 cm2

What is the surface area of this L-shaped prism?

Surface area of a prism

6 cm

5 cm

3 cm

4 cm

3 cm To find the surface area of this shape we need to add together the area of the two L-shapes and the area of the 6 rectangles that make up the surface of the shape.

Total surface area

= 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15= 110 cm2

5 cm

6 cm

3 cm6 cm

3 cm3 cm

3 cm

It can be helpful to use the net of a 3-D shape to calculate its surface area.

Using nets to find surface area

Here is the net of a 3 cm by 5 cm by 6 cm cubiod.

Write down the area of each face.

15 cm2 15 cm2

18 cm2

30 cm2 30 cm2

18 cm2

Then add the areas together to find the surface area.

Surface Area = 126 cm2

Here is the net of a regular tetrahedron.

Using nets to find surface area

What is its surface area?

6 cm

5.2 cm

Area of each face = ½bh

= ½ × 6 × 5.2

= 15.6 cm2

Surface area = 4 × 15.6

= 62.4 cm2

3-Warm up: Finding the Area of a Lateral Face

Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee, are covered with steal panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.

Pyramid Arena

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Surface Area of a ConeSurface Area of a Cone

Unit 6, Day 4

Ms. Reed

With slides from www.cohs.com/.../229_9.3%20Surface%20Area%20of%20Pyramids%20and%20Cones%20C...

A cone has a circular base and a vertex that is not in the same plane as a base.

In a right cone, the height meets the base at its center.

The height of a cone is the perpendicular distance between the vertex and the base.

The slant height of a cone is the distance between the vertex and a point on the base edge.

Height

Lateral Surface

The vertex is directly above the center of the circle.

Baser

Slant Height

r

Surface Area of a Cone Surface Area = area of base + area of sector

= area of base + π(radius of base)(slant height)

S B r 2r r

2B r r

Lateral Area of a Cone

Since Lateral Area = Surface Area – area of the base

2r r L.A. =

Example 1: Find the surface area of the cone to the nearest

whole number.

a. r = 4 slant height = 64 in.

6 in.

2S r r 2(4) (4)(6)

16 24 4040(3.14)

2126 .in

Example 2: Find the surface area of the cone to the nearest whole

number.

b.

First, find the slant height. Next, r = 12,

12 ft.

5 ft.

2 2 2r h 2 2(12) (5)

144 25 169 169 13

13.2S r r

2(12) (12)(13) 144 156 300

2942 .ft

On your own #1

Calculate the surface area of:

•S = (7)2 + (7)(11.40)

•S = 49 + 79.80

•S = 128.8

2S r r

On your own #2Calculate the lateral area of:

•L.A. = (5)(13)

•L.A. = 65

2S r r L.A. =

Homework

Work Packet:

Surface Area of Cones

4-Surface Area of a 4-Surface Area of a SphereSphere

Sphere

Hemisphere

Great Circle

(Surface Area of a Sphere) = 4πr2

5-Basic Geometric 5-Basic Geometric PropertiesProperties

Volume of a cuboid

In this lesson you will learn In this lesson you will learn to calculate the volume of to calculate the volume of

a cuboida cuboid

Cuboids Cuboids

10 cm

4 cm

6 cm

Look at this cuboid

Now imagine it is full of cubic centimetres

Can you see that there are 10 4 = 40 cubic centimetres on the bottom layer?

There are 6 layers of 40 cubes making 40 6 = 240 cm3

1 cm3

10 cm

4 cm

6 cm

Let us go back and look at what we did here

length

breadth

height

When we worked out the volume we multiplied the length by the breadth and then by the heightVolume of a cuboid = length breadth height or

V = l b h

10 cm

4 cm

6 cm

V = l b h

= 10 4 6 cm3

= 240 cm3

Lets us look again at the same cuboid and this time try the formula

You will see that this is the same answer as we got before

6-Volume of a Cylinder6-Volume of a Cylinder

What is Volume?

The volume of a three-dimensional figure is the amount of space within it.

Measured in Units Cubed (e.g. cm3)

Volume of a Prism Volume of a Prism is calculated by

Volume = Area of cross section x perpendicular height

V = Ah

V = (4 x 4) x 4 = 64 m3

What is this?

It has 2 equal shapes at the base, but it is not a prism as it has rounded sides

It is a Cylinder

Volume of a Cylinder

How might we find the Volume of a Cylinder?

Example

V = Ah

Pieces Missing Find the volume of concrete used to make this

pipe Volume of Concrete = Volume of Big

Cylinder – Volume of Small Cylinder (hole)

What shape is present here?

What 3D shapes can you see?

HOME WORK

Find the Volume of the Solid. To 1 decimal place

Homework/Challenge

Challenge Question

Volume of a Cylinder

How might we find the Volume of a Cylinder?

V = Ah– =

Conversion of units

1cm – 10mm 1m – 100cm 1km – 1000m

Conversions of Units1 cm2 = 10 mm x 10 mm =100 mm2

1 m2 = 100 cm x 100 cm = 10 000 cm2

1 m2 = 1000 mm x 1000 mm = 1 000 000 mm2

1 ha = 100 m x 100 m = 10 000 m2

1 km2 = 100 ha

What about when cubic units? 1 cm3

= 1cm x 1cm x 1cm = 10 mm × 10 mm × 10 mm = 1000 mm3

1 m3

= 1m x 1m x 1m = 100 cm × 100 cm × 100 cm = 1 000 000 cm3

Capacity Volume - The volume of a three-dimensional figure is the amount of space within it.

Measured in Units Cubed (e.g. cm3) Volume and capacity are related. Capacity is the amount of material (usually

liquid) that a container can hold. Capacity is measured in millilitres, litres and

kilolitres.

Examples of Capacity

How does Volume relate to Capacity?1000 mL = 1 L

1000 L = 1 kL

1 cm3 = 1 mL

1,000cm3 = 1000ml = 1L

1 m3 = 1000 L = 1 kL

Examples

Convert 1800 mL to L 1800ml = 1800/1000

= 1.8L

2.3 m3 to L 1m3 = 1000L (1kL) 2.3m3 = 2.3kL

= 2300L

Capacity

Find the Capacity of this cube Length = 5.53cm

V = Ah = (5.53 x 5.53) x 5.53 = 169.11cm3 (1cm3 = 1ml)

Capacity = 169.11ml

Length = 5.53cm

Example

Find the capacity of this rectangular prism. Solution Volume = Ah = (26 x 12) x5 = 312 × 5 = 1560 cm3 (1cm3 = 1mL)

Capacity = 1560 mL or 1.56 L (1000mL = 1L)

Ex 11.08 – Q 7.

What size rainwater tank would be needed to hold the run-off when 40 mm of rain falls on a roof 12 m long and 3.6 m wide? (Answer in litres.)

7-Volume of Cones7-Volume of Cones

Volume of Cylinders

Volume = Base x height

V = Bh

Base area = r2 r

h

B

Compare Cone and Cylinder Use plastic space figures. Fill cone with water. Pour water into cylinder. Repeat until cylinder is full.

r r

h

Volume of Cone?

3 cones fill the cylinder, so…

Volume = ⅓ Base x height

=

Volume of Cone

3 cones fill the cylinder Volume = ⅓ Base x height V = ⅓ Bh Base area = r2

V = ⅓ ( . 2.5 2) . 7 V = ⅓ 3.14 . 6.25 . 7 V = 45.79 cm3

r =2.5 cm

h = 7 cm

8-Developing the Formula for the Volume of a Sphere

Volume of a Sphere

Using relational solids and pouring material we noted that the volume of a cone is the same as the volume of a hemisphere (with corresponding dimensions)

Using “math language” Volume (cone) = ½ Volume (sphere)

Therefore 2(Volume (cone)) = Volume (sphere)

=OR +

Volume of a Sphere

We already know the formula for the volume of a cone.

3cylinder

cone

VolumeVolume

= ÷ 3OR

AND we know the formula for the volume of a cylinder

Volume of a Sphere

)() ( HeightXBaseofAreaVolumecylinder

BASE

Hei

ght

SUMMARIZING:

Volume (cylinder) = (Area Base) (height)

Volume (cone) = Volume (cylinder) /3

Volume (cone) = (Area Base) (height)/3

AND 2(Volume (cone)) = Volume (sphere)

Volume of a Sphere

= ÷ 3

2 X =

2(Volume (cone)) = Volume (sphere)

2( ) (height) /3= Volume (sphere)

2( )(h)/3= Volume (sphere)

BUT h = 2r

2(r2)(2r)/3 = Volume(sphere)

4(r3)/3 = Volume(sphere)

Volume of a Sphere

Area of Base

r2

2 X =

hr

r

Volume of a Sphere

3

4 3rVolumesphere

3

4 3r

3

4 3r

3

4 3r

3

4 3r