CST 5 Surface Area Volume

1. Surface AreaSurface Area
and Volumeand Volume

2. Surface Area of PrismsSurface Area of Prisms
Surface AreaSurface Area = The total area of the surface= The total area of the surface
of a three-dimensional objectof a three-dimensional object
(Or think of it as the amount of paper you’ll(Or think of it as the amount of paper you’ll
need to wrap the shape.)need to wrap the shape.)
PrismPrism == A solid object that has two identicalA solid object that has two identical
ends and all flat sides.ends and all flat sides.
E.g. aE.g. a rectangular prismrectangular prism and aand a triangulartriangular
prism.prism.

3. Rectangular
Prism
Triangular
Prism

4. Surface Area (SA)Surface Area (SA)
of a Rectangular Prismof a Rectangular Prism
• Like dice,
there are 6
sides
• or 3 pairs
of sides

5. Prism net - unfoldedPrism net - unfolded

6. • Add the area of all 6 sides to find the
Surface Area..
10 cm - length
5 cm - width
6 cm - height

7. SA = 2 lw + 2 lh + 2 whSA = 2 lw + 2 lh + 2 wh
10 cm - length
5 cm - width
6 cm - height
SA = 2lw + 2lh + 2wh
SA = 2 (10 x 5) + 2 (10 x 6) + 2 (5 x 6)
= 2 (50) + 2(60) + 2(30)
= 100 + 120 + 60
= 280 cm2

8. PracticePractice
10 m
12 m
22 m
SA = 2lw + 2lh + 2wh
= 2(22 x 10) + 2(22 x 12) + 2(10 x 12)
= 2(220) + 2(264) + 2(120)
= 440 + 528 + 240
= 1208 m 2

9. Surface AreaSurface Area
of a Triangular Prismof a Triangular Prism
•2 bases
(triangular)
•3 sides
(rectangular)

10. Net of a Triangular PrismNet of a Triangular Prism

11. Surface Area = 2(Area of triangle)Surface Area = 2(Area of triangle)
+ 3(Area of rectangles)+ 3(Area of rectangles)
15m
Area Triangles = ½ (b x h)
= ½ (12 x 15)
= ½ (180)
= 90
Area Rect. 1 = b x h
= 12 x 25
= 300
Area Rect. 2 = 25 x 20
= 500
SA = 90 + 90 + 300 + 500
+ 500
SA = 1480 m2

12. PracticePractice
10 cm
8 cm
9 cm
7 cm
Triangles = ½ (b x h)
= ½ (8 x 7)
= ½ (56)
= 28 cm
Rectangle 1 = 10 x 8
= 80 cm
Rectangle 2 = 9 x 10
= 90 cm
SA = 28 + 28 + 80 + 90 + 90
SA = 316 cm2

13. Surface Area of a Pop CanSurface Area of a Pop Can

14. ReviewReview
•Surface area is like the amount of
paper you’ll need to wrap the shape.
•You have to “take apart” the shape
and figure the area of the parts.
•Then add them together for the
Surface Area (SA)

15. Parts of a cylinderParts of a cylinder
A cylinder has 2 main
parts.
A rectangle &
A circle – 2 circles really.
Put together they make a
cylinder.

16. The Soup CanThe Soup Can
Think of the cylinder as a soupThink of the cylinder as a soup
can.can.
You have the top and bottom lidYou have the top and bottom lid
((circlescircles) and you have the label) and you have the label
(a(a rectanglerectangle – wrapped around– wrapped around
the can).the can).
The lids and the label are related.The lids and the label are related.
The circumference of the lid is theThe circumference of the lid is the
same as the length of the label.same as the length of the label.

17. Area of the CirclesArea of the Circles
Formula for Area of CircleFormula for Area of Circle
A=A= π rr22
= 3.14 x 3= 3.14 x 322
= 3.14 x 9= 3.14 x 9
= 28.26= 28.26
But there are 2 of them soBut there are 2 of them so
28.26 x 2 = 56.52 units28.26 x 2 = 56.52 units22

18. The Lateral Area RectangleThe Lateral Area Rectangle
This has 2 steps. To find
the area we need base
and height.
Height is given (6) but the
base is not as easy.
Notice that the base is the
same as the distance
around the circle (or the
Circumference).

19. Find CircumferenceFind Circumference
Formula isFormula is
C =C = π x dx d
= 3.14 x 6 (radius doubled)= 3.14 x 6 (radius doubled)
= 18.84= 18.84
Now use that as your base.Now use that as your base.
A = b x hA = b x h
= 18.84 x 6 (the height given)= 18.84 x 6 (the height given)
= 113.04 units= 113.04 units 22

20. Add them togetherAdd them together
Now add the area of theNow add the area of the
circles and the area of thecircles and the area of the
rectangle together.rectangle together.
56.52 + 113.04 = 169.56 units56.52 + 113.04 = 169.56 units22
The total Surface Area!The total Surface Area!

21. Surface Area FormulaSurface Area Formula
SA = (SA = (π d x h) + 2 (d x h) + 2 (π rr22
))
LabelLabel LidsLids
= Area + Area= Area + Area
ofof of 2of 2
Rectangle CirclesRectangle Circles

22. PracticePractice
Be sure you know the difference between a radius and a diameter!Be sure you know the difference between a radius and a diameter!
SA = (SA = (π d x h) + 2 (d x h) + 2 (π rr22
))
= (3.14 x 22 x 14) + 2 (3.14 x 11= (3.14 x 22 x 14) + 2 (3.14 x 1122
))
= (367.12) + 2 (3.14 x 121)= (367.12) + 2 (3.14 x 121)
= (367.12) + 2 (379.94)= (367.12) + 2 (379.94)
= (367.12) + (759.88)= (367.12) + (759.88)
= 1127 cm= 1127 cm22

23. More Practice!More Practice!
SASA = (= (π d x h) + 2 (d x h) + 2 (π rr22
))
= (3.14 x 11 x 7) + 2 ( 3.14 x 5.5= (3.14 x 11 x 7) + 2 ( 3.14 x 5.522
))
= (241.78) + 2 (3.14 x 30.25)= (241.78) + 2 (3.14 x 30.25)
= (241.78) + 2 (3.14 x 94.99)= (241.78) + 2 (3.14 x 94.99)
= (241.78) + 2 (298.27)= (241.78) + 2 (298.27)
= (241.78) + (596.54)= (241.78) + (596.54)
== 838.32 cm838.32 cm22
11 cm
7 cm

24. Surface Area/Pyramid NetsSurface Area/Pyramid Nets
This pyramidThis pyramid
has 2 shapes:has 2 shapes:
1 square &1 square &
4 triangles4 triangles

25. Since you know how to find theSince you know how to find the
areas of those shapes and addareas of those shapes and add
them.them.
Or…Or…

26. you can use a formula…you can use a formula…
SA = ½ l p + B
Where l is the Slant Height and
p is the perimeter and
B is the area of the Base

27. SA = ½ lp + B
6
7
8
5Perimeter = (2 x 7) + (2 x 6) =
26
Slant height l = 8 ;
SA = ½ lp + B
= ½ (8 x 26) + (7 x 6)
*area of the base*
= ½ (208) + (42)
= 104 + 42
= 146 units 2

28. PracticePractice
6
6
18
10SA = ½ lp + B
= ½ (18 x 24) + (6 x 6)
= ½ (432) + (36)
= 216 + 36
= 252 units2
Slant height = 18
Perimeter = 6x4 = 24
What is the extra information in the diagram?

29. Volume of Prisms/CylindersVolume of Prisms/Cylinders
• The number of cubic units
needed to fill the shape.
• Find the volume of this prism by
counting how many cubes tall,
long, and wide the prism is and
then multiplying.
• There are 24 cubes in the prism,
so the volume is 24 cubic units.
2 x 3 x 4 = 24
2 – height
3 – width
4 – length

30. Formula for PrismsFormula for Prisms
VOLUME OF A PRISMVOLUME OF A PRISM
The volumeThe volume VV of a prism is the area of itsof a prism is the area of its
base, Abase, Abb ,, times its height,times its height, hh..
VV == AAbbhh
Note – the capital letter stands for the AREA of the BASENote – the capital letter stands for the AREA of the BASE
not the linear measurement.not the linear measurement.

31. Try ItTry It
4 m -
width
3 m -
height
8 m -
length
V = Abh
Find area of the base
= (8 x 4) x 3
= (32) x 3
Multiply it by the height
= 96 m3

32. PracticePractice
12 cm
10 cm
22 cm
V = A bh
= (22 x10) x 12
= (220) x 12
= 2640 cm3

33. CylindersCylinders
VOLUME OF A CYLINDERVOLUME OF A CYLINDER
The volumeThe volume VV of a cylinder isof a cylinder is
the area of its base,the area of its base, ππrr22
, times its height, times its height hh..
VV == ππrr22
hh
Notice thatNotice that ππrr22
is the formula for area of ais the formula for area of a
circlecircle..

34. Try ItTry It
V = πr2
h
The radius of the cylinder is 5 m,
and the height is 4.2 m
V = 3.14 · 52
· 4.2
V = 329.7
Substitute the
values you know.

35. PracticePractice
7 cm - height
13 cm - radius
V = πr2
h Start with the formula
V = 3.14 x 132
x 7 Substitute what you know
= 3.14 x 169 x 7 Solve using order of Ops.
= 3714.62 cm3

36. Lesson Quiz
Find the volume of each solid to the nearest
tenth. Use 3.14 for π.
861.8 cm34,069.4 m3
312 m3
3. triangular prism: base area = 24 m2
, height = 13 m
1. 2.

37. Volume of Pyramids/ConesVolume of Pyramids/Cones
Remember that Volume of
a Prism is Ab x h, where b
is the area of the base.
You can see that Volume
of a pyramid will be less
than that of a prism.
How much less?

38. Volume of a Pyramid/Cone:
V = (1/3) Area of the Base x height
V = (1/3) Abh
Vol. Pyramid = 1/3 x Vol. Prism
Vol. Cone = 1/3 x Vol. Prism
If you said 2/3 less, you win!
+ + =

39. Find the volume of the square pyramid with
base edge length 9 cm and height 14 cm.
The base is a square with a side
length of 9 cm, and the height
is 14 cm.
V = 1/3 Abh
= 1/3 (9 x 9)(14)
= 1/3 (81)(14)
= 1/3 (1134)
= 378 cm3
14 cm

40. PracticePractice
V = 1/3 A bh
= 1/3 (5 x 5) (10)
= 1/3 (25)(10)
= 1/3 250
= 83.33 units3

41. QuizQuiz
Find the volume of each figure.
1. a rectangular pyramid with length 25
cm, width 17 cm, and height 21 cm
2975 cm3
2. a triangular pyramid with base edge
length 12 cm a base altitude of 9 cm
and height 10 cm.
360 cm3

42. EquivalenceEquivalence
• Equivalence in lines means equal length.Equivalence in lines means equal length.
• Equivalence in polygons means equalEquivalence in polygons means equal
area.area.
• Equivalence in solids means equalEquivalence in solids means equal
volume.volume.
• Page 120, Q, 1, 3Page 120, Q, 1, 3
• Page 133, Q. 1, 2Page 133, Q. 1, 2

43. Equivalent AreasEquivalent Areas
• A triangle is equivalent to a square.A triangle is equivalent to a square.
• What is the side length of the square?What is the side length of the square?
17 cm
20 cm
A = bh/2
A = 20 x 17 /2
A = 170 cm 2
A = side 2
170 = side 2
13 cm = side

44. Equivalent VolumeEquivalent Volume
• A cylinder is equivalent to a cube.A cylinder is equivalent to a cube.
• What is the side length of the cube?What is the side length of the cube?
h = 20 cm
r = 10 cm

45. Perfection of the CanPerfection of the Can
• You know when said you’re not going to walk into a grocery store
and be like “Oh I need to use calculus to get through this shopping
trip!” That’s still going to be true. But that isn’t to say there isn’t
calculus hidden behind everyday objects. For this group problem
set, you’re going do a project analyzing cans.
• Guiding Question:
• How “volume optimized” are the cans in the store?
• Each can in the store is made of a certain amount of metal.
• Could you melt that metal down, re-forge it into a different sized
cylinder which holds even more volume?
• In other words, does that metal enclose the most volume it could?
• What would be the dimension of the best can?