Warm up #3 Page 11draw and label the

shape1. The area of a rectangular rug is 40 yd2. 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 ft2. How many bags will she need to cover the entire garden?

A Prism

Cylinder

Cuboid

Triangular PrismTrapezoid Prism

Volume of Prism = length x Cross-sectional area

Cross section

Area Formulae

Area Circle = πr2

r

Area Rectangle = Base x height

h

b

b

h

Area Triangle = ½ x Base x height

h

bArea Trapezium = ½ x (a + b) x h

a

Geometry

Surface Area of Triangular and cuboid

Prisms

Surface Area

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

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

l

wh

Surface Area

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

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).

7 cm

5 cm2 cm

8 cm

AB

C

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of Sides

Total Area

A

B

C

D

Total

7 cm

5 cm2 cm

8 cm

AB

C

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of

Sides

Total Area

A 40 cm2

1 40 cm2

B

C

D

Total

7 cm

5 cm2 cm

8 cm

AB

C

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of

Sides

Total Area

A 40 cm2

1 40 cm2

B 10 cm2 1 10 cm2

C

D

Total

7 cm

5 cm2 cm

8 cm

AB

C

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of

Sides

Total Area

A 40 cm2

1 40 cm2

B 10 cm2 1 10 cm2

C 35 cm2 1 35 cm2

D

Total

7 cm

5 cm2 cm

8 cm

AB

C

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of

Sides

Total Area

A 40 cm2

1 40 cm2

B 10 cm2 1 10 cm2

C 35 cm2 1 35 cm2

D 7 cm2 2 14 cm2

Total

7 cm

5 cm2 cm

8 cm

AB

CD

Surface Area

We should use a table to tabulate the various areas.

Example: Side Area Number of

Sides

Total Area

A 40 cm2

1 40 cm2

B 10 cm2 1 10 cm2

C 35 cm2 1 35 cm2

D 7 cm2 2 14 cm2

Total 5 99 cm2

7 cm

5 cm2 cm

8 cm

AB

CD

Surface Area

Now you try...find the surface area!

Example:

C

BSide Area No of

SidesArea

2m

11m

2m

2m

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

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

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 cuboid?

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

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

Chequered 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

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.

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

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.

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

B

A

Surface Area

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

A = π r2 Side Area Number of Sides

Total Area

A

B

Totala B =10

5

Surface Area

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

Side Area Number Sides

Total Area

Surface Area

Assignment

Side Area Number Sides

Total Area

AA

4m

2m

Sketch cylinder and copy table. Calculate S.A.

5cm

3cm

Area = π x r2

= π x 32

= π9cm2

Volume = length x Area

= 5 x π9cm2

Volume Cylinder

= 5 x π x 9cm2

=45π

= 45 x π

Lets do these together. Find the volume.

Volume of a CylinderThe volume, V, of a cylinder is V = Bh = r2h, where B is the area of the base, h is the height, and r is the radius of the base.

V = r2h 16

Volume Trapezoid Prismtrapezoid Area = ½ x(a + b) x h

= ½ x (6 + 2) x 5

Volume = length x area

= 20x 4

= 80cm3

2cm4cm

6cm

5cm= ½ x 40cm2

= 20cm2

Volume Trapezoid Prismtrapezoid Area = ½ x(a + b) x h

= ½ x (8 + 3) x 4

Volume = length x area

= 20x 4

= 80cm3

2cm

4cm

8cm

4cm= ½ x cm2

= 20cm2

Geometry

Volume of Rectangular and Triangular

Prisms

VolumeThe same principles apply to

the triangular prism.

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

b

h

Volume

To find the area of the Base…

Area (triangle) = b x h 2

This gives us the Area of the Base (B).b

h

Volume

Now to find the volume…

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.

B h

Volume

Volume of a Triangular Prism

Volume (triangular prism)

V = B x hB h

Volume

Together…Volume

V = B x h

Volume

Together…Volume

V = B x h

V = (8 x 4) x 12 2

Volume

Together…Volume

V = B x h

V = (8 x 4) x 12 2V = 16 x 12

Volume

Together…Volume

V = B x h

V = (8 x 4) x 12 2V = 16 x 12

V = 192 cm3

Volume

Your turn… 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 PrismCross-sectional Area = ½ x b x h

= ½ x 8 x 4

Volume = length x CSA

= 16 x 6

= 96cm3

8cm6cm

4cm 4.9cm= .5 x 32

= 16cm2

Find the Volume of the Triangular Prism.

24862

1 Base Triangular of Area

6

8

!!

4

10

4

10

2401024 height x Base

Volume Cuboid

5cm

7cm

10cm

Cross-sectional Area = b x h

= 7 x 5= 35cm2

Volume = length x CSA

= 10 x 35

= 350cm3

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

1cm

How many 1cm3 cubes will fill the rectangular prism on the right

Volume of a three-dimensional figure is the number of cubic units needed to fill the space inside the figure.

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

6

710

Find the volume. BlwV 107B(base)

70BBhV hV 98

670V 588V

Volume of a Cube The volume of a cube is the length of its side cubed, or V=s3

9 in.

9 in.

9 in.

Find the volume. V=s3

39V999 V

3inches 729V

Volume of a cuboid

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

Volume of a cuboid

= length × width × height

= lwh

height, h

length, lwidth, w

The area of the base

= length × width

So,

Volume of a cuboid

What is the volume of this cuboid?

Volume of cuboid

= length × width × height

= 5 × 8 × 13

= 520 cm3

5 cm

8 cm 13 cm

What is the volume of this L-shaped prism?

Volume of a prism made from cuboids

6 cm

5 cm

3 cm

4 cm

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

Volume of the green cuboid

= 6 × 3 × 3 = 54 cm3

Volume of the blue cuboid

= 3 × 2 × 2 = 12 cm3

Total volume

= 54 + 12 = 66 cm3

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

Volume of a prism

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 cm2 and the length is 3 cm,

Volume of L-shaped prism = 22 × 3 = 66 cm3

3 cm

Volume of a prism

Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 =

Volume of prism = 5 × 72 = 360 m3

3 m

4 m

12 m

7 m

5 m

72 m2

What is the volume of this prism?