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Surface Surface Area and Area and VolumeVolume
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Surface Area of PrismsSurface Area of PrismsSurface AreaSurface Area = The total area of the = The total area of the
surface of a three-dimensional object surface of 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 A solid object that has two identical ends and all flat sides.identical ends and all flat sides.
We will start with 2 prisms – a We will start with 2 prisms – a rectangular rectangular prismprism and a and a triangular prism.triangular prism.
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Rectangular Prism
Triangular Prism
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Surface Area (SA) of a Surface Area (SA) of a Rectangular PrismRectangular Prism
Like dice, there are six sides (or 3 pairs of sides)
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Prism net - unfoldedPrism net - unfolded
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• Add the area of all 6 sides to find the Add the area of all 6 sides to find the Surface Area.Surface Area.
10 - length5 - width
6 - height
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SA = 2lw + 2lh + 2whSA = 2lw + 2lh + 2wh
10 - length
5 - width
6 - 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 units squared
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PracticePractice
10 ft
12 ft
22 ft
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 ft squared
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Surface Area of a Surface Area of a Triangular PrismTriangular Prism
•2 bases (triangular)
•3 sides (rectangular)
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Unfolded net of a triangular Unfolded net of a triangular prismprism
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2(area of triangle) + Area 2(area of triangle) + Area of rectanglesof rectangles
15ft
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 ft squared
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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
Add them all up
SA = 28 + 28 + 80 + 90 + 90
SA = 316 cm squared
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Surface Area of a Surface Area of a CylinderCylinder
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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)
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Parts of a cylinderParts of a cylinderA cylinder has 2 main parts.
A rectangle
and
A circle – well, 2 circles really.
Put together they make a cylinder.
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The Soup CanThe Soup CanThink of the Cylinder as a soup Think of the Cylinder as a soup
can. can.
You have the top and bottom lid You 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 The circumference of the lid is the same as the length of the the same as the length of the label. label.
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Area of the CirclesArea of the Circles
Formula for Area of CircleFormula for Area of Circle
A= A= r r2 2
= 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 units 28.26 x 2 = 56.52 units squaredsquared
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The RectangleThe 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).
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Find CircumferenceFind Circumference
Formula is Formula is C = C = x d x 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 = 18.84 x 6 (the height given)given)= 113.04 units squared= 113.04 units squared
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Add them togetherAdd them together
Now add the area of the Now add the area of the circles and the area of circles and the area of the rectangle together.the rectangle together.
56.52 + 113.04 = 169.56 56.52 + 113.04 = 169.56 units squared units squared
The total Surface Area!The total Surface Area!
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FormulaFormula
SA = (SA = ( d x h) + 2 ( d x h) + 2 ( r r22))
LabelLabel Lids Lids (2)(2)
Area of Rectangle Area Area of Rectangle Area of Circlesof Circles
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PracticePracticeBe sure you know the difference between a radius and a Be sure you know the difference between a radius and a
diameter!diameter!
SA = (SA = ( d x h) + 2 ( d x h) + 2 ( r r22))
= (3.14 x 22 x 14) + 2 (3.14 = (3.14 x 22 x 14) + 2 (3.14 x 11x 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
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More Practice!More Practice!SA SA = (= ( d x h) + 2 ( d x h) + 2 ( r r22))
= (3.14 x 11 x 7) + 2 ( 3.14 = (3.14 x 11 x 7) + 2 ( 3.14 x 5.5x 5.522))
= (241.78) + 2 (3.14 x = (241.78) + 2 (3.14 x 30.25)30.25)
= (241.78) + 2 (3.14 x = (241.78) + 2 (3.14 x 94.99)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
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Surface Area of a Surface Area of a PyramidPyramid
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Pyramid NetsPyramid Nets
A pyramid has A pyramid has 2 shapes:2 shapes:
One (1) One (1) square square
&&
Four (4) Four (4) trianglestriangles
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Since you know how to find Since you know how to find the areas of those shapes the areas of those shapes
and add them.and add them.
Or…Or…
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you can use a formula…you can use a formula…
SA = ½ lp + B
Where l is the Slant Height and
p is the perimeter andB is the area of the Base
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SA = ½ lp + B
6
7
8
5Perimeter = (2 x 7) + (2 x 6) = 26Slant 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
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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?
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Volume of Prisms and Volume of Prisms and CylindersCylinders
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VolumeVolume• 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
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Formula for PrismsFormula for Prisms
VOLUME OF A PRISMVOLUME OF A PRISM
The volume The volume VV of a prism is the of a prism is the area of its base area of its base BB times its height times its height hh..
VV = = BhBh
Note – the capital letter stands for the AREA of the Note – the capital letter stands for the AREA of the BASE not the linear measurement. BASE not the linear measurement.
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Try ItTry It
4 ft - width
3 ft - height
8 ft - length
V = BhFind area of the base
= (8 x 4) x 3
= (32) x 3Multiply it by the height
= 96 ft3
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PracticePractice
12 cm
10 cm
22 cm
V = Bh= (22 x 10)
x 12= (220) x
12= 2640 cm3
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CylindersCylinders
VOLUME OF A CYLINDERVOLUME OF A CYLINDER
The volume The volume VV of a cylinder is the area of a cylinder is the area of its base, of its base, rr22, times its height , times its height hh..
VV = = rr22hh
Notice that Notice that rr2 2 is the formula for area is the formula for area of a circle.of a circle.
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Try ItTry It
V = r2h
The radius of the cylinder is 5 m, and the height is 4.2 mV = 3.14 · 52 · 4.2
V = 329.7
Substitute the values you know.
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PracticePractice
7 cm - height
13 cm - radius
V = r2h 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
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Lesson Quiz
Find the volume of each solid to the nearest tenth. Use 3.14 for .
861.8 cm34,069.4 m3
312 ft3
3. triangular prism: base area = 24 ft2, height = 13 ft
1. 2.
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Volume of PyramidsVolume of Pyramids
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Remember that Volume of a Prism is B 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? Any guesses?
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Volume of a Pyramid:V = (1/3) Area of the Base x heightV = (1/3) BhVolume of a Pyramid = 1/3 x Volume of a Prism
If you said 2/3 less, you win!
+ + =
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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 Bh= 1/3 (9 x 9)(14)= 1/3 (81)(14)= 1/3 (1134)= 378 cm3
14 cm
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PracticePractice
V = 1/3 Bh= 1/3 (5 x 5) (10)= 1/3 (25)(10)= 1/3 250= 83.33 units3
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Try This:Try This:Find the volume of each figure.
1. a rectangular pyramid with length 25 cm, width 17 cm, and height 21 cm2975 cm3
2. a triangular pyramid with base edge length 12 in. a base altitude of 9 in. and height 10 in.360 in3