Warm up #3 Page 11

draw and label the shape

1. The area of a rectangular rug is 40 yd 2 . If the width of the rug is 10 yd, what is the length of the rug? 2. The perimeter of a square rug is 16yd. If the width of the rug is 4 yd, what is the length of the rug?

3. Jose wants new carpeting for his living room. His living room is an 9 m by 9 m rectangle. How much carpeting does he need to buy to cover his entire living room?

4. Patricia has a rectangular flower garden that is 10 ft long and 5 ft wide. One bag of soil can cover 10 ft 2 . How many bags will she need to cover the entire garden?

Cylinder

A Prism

Cuboid Cross section Triangular Prism Trapezoid Prism Volume of Prism = length x Cross-sectional area

r Area Circle = πr 2 a h b Area Trapezium = ½ x (a + b) x h

Area Formulae

h b Area Rectangle = Base x height h b Area Triangle = ½ x Base x height

Geometry

Surface Area of Triangular and cuboid Prisms

Surface Area



Triangular prism – a prism with two parallel, equal triangles on opposite sides.

h l w

To find the surface area of a triangular prism we can add up the areas of the separate faces.

Surface Area



In a triangular prism there are two pairs of opposite and equal triangles.

A

2 cm

B C

7 cm 8 cm 5 cm

We can find the surface area of this prism by adding the areas of the pink side (A), the orange sides (B), the green bottom (C) and the two ends (D).

Surface Area



We should use a table to tabulate the various areas.

Example: A

2 cm

B C

7 cm 8 cm 5 cm Side A B C D Total Area Number of Sides Total Area

Surface Area



We should use a table to tabulate the various areas.

Example: A

2 cm

B C

7 cm 8 cm 5 cm Side A B C D Total Area 40 cm 2 Number of Sides 1 Total Area 40 cm 2

Surface Area



We should use a table to tabulate the various areas.

Example: A

2 cm

B C

7 cm 8 cm 5 cm Side A B C D Total Area 40 cm 2 10 cm 2 Number of Sides 1 Total Area 40 cm 2 1 10 cm 2

Surface Area



We should use a table to tabulate the various areas.

Example: A

2 cm

B C

7 cm 8 cm 5 cm Side A B C D Total Area 40 cm 2 10 cm 2 35 cm 2 Number of Sides 1 Total Area 40 cm 2 1 1 10 35 cm cm 2 2

Surface Area



We should use a table to tabulate the various areas.

Example:

2 cm

A

B

D

C

7 cm 8 cm 5 cm Side A B C D Total Area 40 cm 2 10 cm 2 35 cm 2 7 cm 2 Number of Sides 1 Total Area 40 cm 2 1 1 2 10 35 14 cm cm cm 2 2 2

Surface Area



We should use a table to tabulate the various areas.

Example:

2 cm

A

B

D

C

7 cm 8 cm 5 cm Side A B C D Total Area 40 cm 2 10 cm 2 35 cm 2 7 cm 2 Number of Sides 1 Total Area 40 cm 2 1 1 2 5 10 35 14 99 cm cm cm cm 2 2 2 2

Surface Area

Example:



Now you try...find the surface area!

B

Side Area No of Sides Area

2m C 2m 11m 2m

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 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.

Formula for the surface area of a cuboid

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

Surface area of a cuboid =

w l 2

×

lw Top and bottom h + 2

×

hw Front and back + 2

×

lh Left and right side

= 2 lw + 2 hw + 2 lh

Surface area of a cuboid

To find the

surface area

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

8 cm 5 cm Can you work out the surface area of this cuboid?

7 cm The area of the top = 8

×

5 = 40 cm 2 The area of the front = 7

×

5 = 35 cm 2 The area of the side = 7

×

8 = 56 cm 2

Surface area of a cuboid

To find the

surface area

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

8 cm 5 cm So the total surface area = 2

×

40 cm 2 Top and bottom 7 cm + 2

×

35 cm 2 Front and back + 2

×

56 cm 2 Left and right side = 80 + 70 + 112 =

262 cm 2

Chequered cuboid problem

This cuboid is made from alternate purple and green centimetre cubes. What is its surface area?

Surface area =

2

×

3

×

4 + 2

×

3

×

5 + 2

×

4

×

5

= 24 + 30 + 40 =

94 cm 2

How much of the surface area is green?

48 cm 2

Surface area of a prism

6 cm 3 cm What is the surface area of this L-shaped prism?

3 cm 4 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.

5 cm Total surface area = 2

×

22 + 18 + 9 + 12 + 6 + 6 + 15 =

110 cm 2

Using nets to find surface area

Here is the net of a 3 cm by 5 cm by 6 cm cuboid Write down the area of each face.

6 cm Then add the areas together to find the surface area.

3 cm

18 cm 2

3 cm 6 cm 5 cm

15 cm 2 30 cm 2 15 cm 2 30 cm 2

3 cm

18 cm 2

3 cm Surface Area =

126 cm 2

Surface Area



Cylinder – (circular prism) a prism with two parallel, equal circles on opposite sides.

To find the surface area of a cylinder we can add up the areas of the separate faces.

Surface Area



In a cylinder there are a pair of opposite and equal circles.

A

B

We can find the surface area of a cylinder by adding the areas of the two blue ends (A) and the yellow sides (B).

Surface Area



We can find the area of the two ends (A) by using the formula for the area of a circle.



A = π r 2 Side Area Number of Sides Total Area A B Total

a

5



Sketch cylinder and copy table. Work together to find the S.A.

Side

Surface Area

Area Number Sides Total Area



Assignment



Sketch cylinder and copy table. Calculate S.A.

Side Area

Surface Area

Number Sides Total Area 4m AA

Volume Cylinder

Area = π x r 2 = π x 3 2 = π9cm 2 3cm 5cm Volume = length x Area = 5 x π9cm 2 = 5 x

π x 9cm

2 = 45 x

π

=45

π

Lets do these together. Find the volume.

V =



r 2 h 16

Volume of a Cylinder The volume, V, of a cylinder is V = Bh =



r 2 h, where B is

the area of the base, h is the height, and r is the radius of

the base.

Volume Trapezoid Prism

trapezoid Area = ½ x(a + b) x h = ½ x (6 + 2) x 5

= ½ x 40cm 2 = 20cm 2

6cm 5cm 4cm 2cm Volume = length x area = 20x 4 = 80cm 3

Volume Trapezoid Prism

trapezoid Area = ½ x(a + b) x h = ½ x (8 + 3) x 4

= ½ x cm 2 = 20cm 2

8cm 4cm 4cm 2cm Volume = length x area = 20x 4 = 80cm 3

Geometry

Volume of Rectangular and Triangular Prisms



The same principles apply to the triangular prism.

Volume

h b

To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow).

Volume



To find the area of the Base…

h b

Area (triangle) = b x h 2 This gives us the Area of the Base (B).

Volume



Now to find the volume…

B h We must then multiply the area of the base (B) by the height (h) of the prism.

This will give us the Volume of the Prism.

Volume



Volume of a Triangular Prism

B h Volume (triangular prism) V = B x h



Together…

Volume V = B x h

Volume

Volume



Together…

Volume V = B x h V = (8 x 4) 2 x 12

Volume



Together…

Volume V = B x h V = (8 x 4) 2 V = 16 x 12 x 12

Volume



Together…

Volume V = B x h V = (8 x 4) 2 V = 16 x 12 x 12 V = 192 cm 3



Your turn…

Volume

Find the Volume

Triangular Prism

 To find the

volume

of a triangular prism find the area of the triangular base and multiply times the height of the prism. The height will always be the distance between the two triangles.

Volume Triangular Prism

Cross-sectional Area = ½ x b x h = ½ x 8 x 4 = .5 x 32 = 16 cm 2 4cm 4.9cm

8cm Volume = length x CSA = 16 x 6 = 96cm 3 6cm

Find the Volume of the Triangular Prism.

Area of Triangular Base  1 2  24

4 10 !

!

8 4 6 10

Base x height



24



10



240

Volume Cuboid

Cross-sectional Area = b x h = 7 x 5 = 35cm 2 5cm 10cm 7cm Volume = length x CSA = 10 x 35 = 350cm 3

Ex. 1: Finding the Volume of a rectangular prism

 The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box?

VOLUMES OF PRISMS AND CYLINDERS Volume of a three-dimensional figure is the number of cubic units needed to fill the space inside the figure.

1cm How many 1cm 3 cubes will fill the rectangular prism on the right

Find the volume.

6 7 10 V V



Blw



B(base)

B

 70

Bh V



7



10

 98 

h V

 70  6

V

 588

Volume of a Prism The volume, V, of a prism is V = Bh, where B is the area of the base and h is the height.

Find the volume.

9 in.

9 in.

9 in.

V=s 3

V V V

  9 9  3 9  9 

729 inches

3

Volume of a Cube The volume of a cube is the length of its side cubed, or V=s 3

Volume of a cuboid

We can find the volume of a cuboid by multiplying the area of the base by the height.

The area of the base = length

×

width So, height , h Volume of a cuboid = length

×

width

×

height = lwh length , l width , w

Volume of a cuboid

5 cm 8 cm What is the volume of this cuboid?

13 cm Volume of cuboid = length

×

width

×

height = 5

×

8

×

13 =

520 cm 3

Volume of a prism made from cuboids

6 cm 3 cm What is the volume of this L-shaped prism?

3 cm We can think of the shape as two cuboids joined together.

4 cm Volume of the green cuboid = 6

×

3

×

3 = 54 cm 3 5 cm Volume of the blue cuboid = 3

×

2

×

2 = 12 cm 3 Total volume = 54 + 12 =

66 cm 3

Volume of a prism

Remember, a prism is a 3-D shape with the same cross-section throughout its length.

3 cm We can think of this prism as lots of L-shaped surfaces running along the length of the shape.

Volume of a prism = area of cross-section

×

length If the cross-section has an area of 22 cm 2 and the length is 3 cm, Volume of L-shaped prism = 22

×

3 =

66 cm 3

Volume of a prism

What is the volume of this prism?

7 m 4 m 12 m 3 m 5 m Area of cross-section = 7

×

12 – 4

×

3 = 84 – 12 = Volume of prism = 5

×

72 =

360 m 3

72 m 2