Warm up #3 Page 11
draw and label the shape
1. 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
Cross section
Trapezoid Prism
Triangular Prism
Volume of Prism = length x Cross-sectional area
Area Formulae
r
h
Area Circle = πr2
a
b
Area Rectangle
= Base x height
h
b
Area Trapezium
= ½ x (a + b) x h
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
w
l
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.
8 cm
A
2 cm B C
7 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:
Side
8 cm
A
2 cm B C
7 cm
5 cm
A
B
C
D
Total
Area
Number
of Sides
Total
Area
Surface Area
 We should use a table to tabulate the
various areas.
Example:
8 cm
A
2 cm B C
7 cm
5 cm
Side
Area
Number
of Sides
Total
Area
A
40 cm2
1
40 cm2
B
C
D
Total
Surface Area
 We should use a table to tabulate the
various areas.
Example:
8 cm
A
2 cm B C
7 cm
5 cm
Side
Area
Number
of Sides
Total
Area
A
40 cm2
1
40 cm2
B
10 cm2
1
10 cm2
C
D
Total
Surface Area
 We should use a table to tabulate the
various areas.
Example:
8 cm
A
2 cm B C
7 cm
5 cm
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
Surface Area
 We should use a table to tabulate the
various areas.
Example:
8 cm
A
2 cm BD C
7 cm
5 cm
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
Surface Area
 We should use a table to tabulate the
various areas.
Example:
8 cm
A
2 cm BD C
7 cm
5 cm
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
5
99 cm2
Total
Surface Area
Example:
 Now you try...find the surface area!
B
Side
Area
No of
Sides
2m
C
2m
11m
2m
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 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 =
l
h
w
2 × lw
Top and bottom
+ 2 × hw
Front and back
+ 2 × lh
Left and right side
= 2lw + 2hw + 2lh
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
5 cm
surface area of this cuboid?
8 cm
The area of the top = 8 × 5
= 40 cm2
7 cm
The area of the front = 7 × 5
= 35 cm2
The area of the side = 7 × 8
= 56 cm2
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 cm2
7 cm
Top and bottom
+ 2 × 35 cm2 Front and back
+ 2 × 56 cm2 Left and right side
= 80 + 70 + 112 = 262 cm2
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 cm2
How much of the
surface area is green?
48 cm2
Surface area of a prism
What is the surface area of this L-shaped prism?
3 cm
3 cm
4 cm
6 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
5 cm
= 2 × 22 + 18 + 9 + 12 + 6
+ 6 + 15
= 110 cm2
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
3 cm
18 cm2
3 cm
5 cm 15 cm2
30 cm2
15 cm2
3 cm
18 cm2
3 cm
Then add the
areas together
to find the
surface area.
6 cm
30 cm2
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.
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
2
r
Side
A
B
a
5
Total
Area
Number
of Sides
Total
Area
 Sketch cylinder and
Surface Area
copy table. Work
together to find the
S.A.
Side
Area
Number
Sides
Total
Area
 Assignment
Surface Area
 Sketch cylinder and copy
table. Calculate S.A.
Side
4m
AA
Area
Number
Sides
Total
Area
Volume Cylinder
Area = π x r2
= π x 32
= π9cm2
3cm
5cm
Volume = length x Area
= 5 x π9cm2
= 5 x π x 9cm2
= 45 x π
=45π
Lets do these together. Find the
volume.
V = r h
2
16
Volume of a Cylinder
The 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.
Volume Trapezoid Prism
trapezoid Area = ½ x(a + b) x h
= ½ x (6 + 2) x 5
= ½ x 40cm2
6cm
5cm
= 20cm2
4cm
2cm
Volume = length x area
= 20x 4
= 80cm3
Volume Trapezoid Prism
trapezoid Area = ½ x(a + b) x h
= ½ x (8 + 3) x 4
=½x
cm2
8cm
4cm
= 20cm2
4cm
2cm
Volume = length x area
= 20x 4
= 80cm3
Geometry
Volume of
Rectangular and Triangular
Prisms
 The same principles apply to the
triangular prism.
h
b
Volume
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…
Area (triangle) = b x h
2
h
b
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
Volume
(triangular prism)
B
h
V =
B x h
Volume
 Together…
Volume
V =
B x h
Volume
 Together…
Volume
V =
V=
B x h
(8 x 4) x 12
2
Volume
 Together…
Volume
V = B
x
h
V = (8 x 4) x 12
2
V = 16
x 12
Volume
 Together…
Volume
V = B x h
V = (8 x 4) x 12
2
V = 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 Prism
Cross-sectional Area = ½ x b x h
=½x8x4
= .5 x 32
4cm
4.9cm
= 16cm2
6cm
8cm
Volume = length x CSA
= 16 x 6
= 96cm3
Find the Volume of the
Triangular Prism.
1
Area of Triangular Base  6  8  24
2
4
10
4
10
!
!
8
Base x height  24 10  240
6
Volume Cuboid
Cross-sectional Area = b x h
=7x5
= 35cm2
5cm
10cm
7cm
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
Volume of a three-dimensional figure is the number
of cubic units needed to fill the space inside the
figure.
1cm
How many 1cm3 cubes will fill the
rectangular prism on the right
Find the volume.
10
7
6
V  Blw
B(base)  7 10
B  70
V  Bh
V  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.
3
V=s
V 9
9 in.
9 in.
3
V  999
V  729inches
3
9 in.
Volume of a Cube
The volume of a cube is the length of its side cubed,
or V=s3
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
width, w
length, l
Volume of a cuboid
What is the volume of this cuboid?
Volume of cuboid
= length × width × height
5 cm
= 5 × 8 × 13
8 cm
13 cm
= 520 cm3
Volume of a prism made from cuboids
What is the volume of this L-shaped prism?
3 cm
We can think of the shape as
two cuboids joined together.
3 cm
4 cm
Volume of the green cuboid
= 6 × 3 × 3 = 54 cm3
6 cm
Volume of the blue cuboid
= 3 × 2 × 2 = 12 cm3
Total volume
5 cm
= 54 + 12 = 66 cm3
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 cm2 and the length is 3 cm,
Volume of L-shaped prism = 22 × 3 =66 cm3
Volume of a prism
What is the volume of this prism?
12 m
7m
4m
3m
5m
Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 m2
Volume of prism = 5 × 72 =360 m3