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? - PowerPoint PPT Presentation
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?
Transcript
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 PrismCylinder
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.
lwh
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 cm5 cm2 cm
8 cmA
B C
Surface Area
We should use a table to tabulate the various areas.
Example: Side Area Number of Sides
Total Area
ABCD
Total
7 cm5 cm2 cm
8 cmA
B 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
BCD
Total
7 cm5 cm2 cm
8 cmA
B 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
CD
Total
7 cm5 cm2 cm
8 cmA
B 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
DTotal
7 cm5 cm2 cm
8 cmA
B 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 cm5 cm2 cm
8 cmA
B 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 cm5 cm2 cm
8 cmA
B 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
l w
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 cm 5 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 cm 5 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 areaHere 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
4m2m
Sketch cylinder and copy table. Calculate S.A.
5cm3cm
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
2cm4cm
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.
248621 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
