surface area of a cuboid and a cube,surface area of a right circular cylinder,surface area of right circular cone,surface area of a sphere,volume of cuboid,volume of cylinder,volume of right circular cone and volume of sphere.powerpoint presentation
3. Objectives
At the end of the lesson the students
should be able;
To find the surface area of a
cylinder ..
4. 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.
6. What will happen if we
removed the end of the
cylinder and unrolled the
body?
Lets find out
!!!!
7. This will happen if we unrolled
and removed the end of a
cylinder….
h
Circumference
of the base
2Πr 2
8. 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
9. 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.
10. 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
11. We derived at this formula..!!
SA=2Πr2 +2Πr
Or
SA=2Πr (r + h)
12. 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:
SA=2πr2 +2πrh
h=20m
r=5m =2(3.14)(5m)2 + 2[(3.14)
(5m)(20m)
=157m2 + 628m
SA =785m2
14. 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.
15. 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.
16. 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.
17. 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
8 cm surface area of this cubiod?
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
18. Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
5 cm So the total surface area =
8 cm
2 × 40 cm2 Top and bottom
7 cm + 2 × 35 cm2 Front and back
+ 2 × 56 cm2 Left and right side
= 80 + 70 + 112 = 262 cm2
19. 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
= 2lw + 2hw + 2lh
20. Surface area of a cube
How can we find the surface area of a cube of length x?
All six faces of a cube have the
same area.
The area of each face is x × x = x2
Therefore,
x
Surface area of a cube = 6x2
21. Checkered 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
22. Surface area of a prism
What is the surface area of this L-shaped prism?
3 cm
To find the surface area of
3 cm
this shape we need to add
together the area of the two
4 cm L-shapes and the area of the
6 rectangles that make up
6 cm the surface of the shape.
Total surface area
= 2 × 22 + 18 + 9 + 12 + 6
+ 6 + 15
5 cm = 110 cm2
23. Using nets to find surface area
It can be helpful to use the net of a 3-D shape to calculate its
surface area.
Here is the net of a 3 cm by 5 cm by 6 cm cubiod.
6 cm
Write down the
area of each
3 cm 18 cm2 3 cm
6 cm face.
Then add the
5 cm 15 cm2 30 cm2 15 cm2 30 cm2
areas together
to find the
surface area.
3 cm 18 cm2 3 cm
Surface Area = 126 cm2
24. Using nets to find surface area
Here is the net of a regular tetrahedron.
What is its surface area?
Area of each face = ½bh
= ½ × 6 × 5.2
= 15.6 cm2
5.2 cm Surface area = 4 × 15.6
= 62.4 cm2
6 cm
25. 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.
28. Surface 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...
29. 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 vertex is directly
Height above the center of
the circle.
Lateral Surface
Slant Height
r
Base
r
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.
30. Surface Area of a Cone
Surface Area = area of base + area of sector
= area of base + π(radius of base)(slant height)
S = B + π r l = π r + π rl 2
l
B =πr 2 r
31. Lateral Area of a Cone
Since Lateral Area = Surface Area – area of the
base
= π r + π rl
L.A. = 2
32. Example 1:
Find the surface area of the cone to the nearest
whole number.
a. 4 in. r = 4 slant height = 6
S = π r + π rl
2
= π (4) + π (4)(6)
2
6 in.
= 16π + 24π
= 40π
= 40(3.14)
≈ 126in. 2
33. Example 2:
Find the surface area of the cone to the nearest whole
number.
b. l
5 ft.
12 ft.
First, find the slant height. Next, r = 12, l = 13.
l =r +h
2 2 2 S = π r + π rl
2
= (12) + (5)
2 2 = π (12) + π (12)(13)
2
= 144 + 25 = 169 = 144π + 156π
= 300π
l = 169 = 13 ≈ 942 ft. 2
34. On your own #1
Calculate the surface
area of:
S = π r 2 + π rl
•S = π(7)2 + π(7)(11.40)
•S = 49π + 79.80π
•S = 128.8π
35. On your own #2
Calculate the lateral area of:
S = π r = π rl
L.A. +
2
•L.A. = π(5)(13)
•L.A. = 65π
47. Look at this cuboid
Now imagine it is full
of cubic centimetres
6 cm
1 cm3
4 cm
10 cm
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
48. Let us go back and look at what we did here
6 cm
height
4 cm
breadth
10 cm
length
When we worked out the volume we multiplied the length by the
breadth and then by the height
Volume of a cuboid = length × breadth × height
or
V=lbh
49. Lets us look again
at the same
cuboid and this 6 cm
time try the
formula
4 cm
10 cm
V=lbh
= 10 × 4 × 6 cm3
= 240 cm3
You will see that this is the same answer as we got before
56. Pieces Missing
Find the volume of concrete used to make this
pipe
Volume of Concrete = Volume of Big
Cylinder – Volume of Small Cylinder (hole)
63. Conversions of Units
1 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
64. 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
65. 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.
75. Compare Cone and Cylinder
Use plastic space figures.
Fill cone with water.
Pour water into cylinder.
Repeat until cylinder is full.
r r
h
76. Volume of Cone?
=
3 cones fill the cylinder, so…
Volume = ⅓ Base x height
77. Volume of Cone
3 cones fill the cylinder
Volume = ⅓ Base x height
V = ⅓ Bh h = 7 cm
Base area = π r2
V = ⅓ (π . 2.5 2) . 7 r =2.5 cm
V = ⅓ 3.14 . 6.25 . 7
79. 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 =
+
80. Volume of a Sphere
We already know the formula for the volume of a cone.
Volumecylinder
Volumecone =
3
OR = ÷3
81. Volume of a Sphere
AND we know the formula for the volume of a cylinder
Volumecylinder = ( Area of Base ) X (Height )
Height
BASE
83. Volume of a Sphere
2(Volume (cone)) = Volume (sphere)
2X =
2(Area of Base) (height) /3= Volume (sphere)
2( πr2)(h)/3= Volume (sphere)
r
BUT h = 2r h
r
2(πr )(2r)/3 = Volume(sphere)
2
4(πr3)/3 = Volume(sphere)
84. 3
4π r
Volume of a Sphere 4 π r 3
3 3
3
4π r
Volumesphere =
3
4 π r 3 4 π r 3
3 3
Hinweis der Redaktion
Discuss the meaning of surface area. The important thing to remember is that although surface area is found for three-dimensional shapes, surface area only has two dimensions. It is therefore measured in square units.
Stress the importance to work systematically when finding the surface area to ensure that no faces have been left out. We can also work out the surface area of a cuboid by drawing its net ( see slide 51 ). This may be easier for some pupils because they would be able to see every face rather than visualizing it.
Pupils should write this formula down.
As pupils to use this formula to find the surface area of a cube of side length 5 cm. 6 × 5 2 = 6 × 25 = 150 cm 2 . Repeat for other numbers. As a more challenging question tell pupils that a cube has a surface area of 96 cm 2 . Ask them how we could work out its side length using inverse operations.
Discuss how to work out the surface area that is green. Ask pupils how we could write the proportion of the surface area that is green as a fraction, as a decimal and as a percentage.
Discuss ways to find the surface area of this solid. We could use a net of this prism to help find the area of each face.
Links: S3 3-D shapes – nets S6 Construction and Loci – constructing nets