Surface Area & Volume
Prism & Cylinders
• A polyhedron (3D shape) has faces that are polygons (2D shape)- Poly means many; hedron means faces
• Two faces meet at an edge• Three or more edges meet at a vertex
Edge
Vertex
A net is a two-dimensional figure that, when folded, forms a three-dimensional figure
Nets of a Cube
• Prism: A solid object that has two identical ends (bases) and all flat sides
Bases
Triangular Prism
Height
Triangular prismRectangular prism
Base
Height
Base
Prism Names
Name the prism
Side 2
Bottom
Back
Top
Side 1Front
Base 2
Bottom
Back
Top
Base 1Front
Length (L)Base (B)
Height (H)
Rectangular Prism
Example: Rectangular Prism Net
This is a rectangular prism. The l, w and h have different values.l
w
h
Surface Area - the total area of all the faces (surfaces) of an object.
To find the surface area of an object we can add up the areas of the separate faces. Note: some faces have the same size and shape
C
D
A
Surface Area of this prism:
Add the areas of the two brown sides (A) to the two green sides (D) and to the two red sides (C).
Surface Area
• Find the surface area of the rectangular prism• We should use a table to tabulate the various areas.
Face Area Number of
Sides
Total Area
A 12cm2 2 24cm2
D 15cm2 2 30cm2
C 20cm2 2 40cm2
TOTAL
6 94cm2
Area of rectangle = l x w
3 cm5 cm
CD
A
4 cm
Volume
• The number of cubic units needed to fill the space occupied by a solid.
2 cm 5 cm
3 cm
Volume = Base area x height
= (l x w) x height
= (3 cm x 2 cm) x height
= 6 cm2 x 5 cm
= 30 cm3
VOLUME OF A PRISM
The volume V of a prism is the area of its base B times its height h.
V = BhNote – the capital letter stands for
the AREA of the BASE not the linear measurement.
B B Bh h h
Area of rectangle = l x w
Prism Volume Example
4 ft
3 ft
8 ft
V = BhFind area of the base 1st
= (4ft x 3ft)h= (12 ft2)h
Multiply it by the height
= (12 ft2) x 8 ft= 96 ft3
B
Area of rectangle = l x w
What is the surface area and volume of the prism?
12 cm
10 cm
22 cm
V = Bh= (12 cm x 10 cm) x h= (120 cm2) x 22 cm= 2640 cm3
SA rectangle = l x w
SA (rect1) = 12 cm x 10 cm = 120 cm2 x 2 = 240 cm2
SA (rect2) = 12 cm2 x 22 cm= 264 cm2 x 2 = 528 cm2
SA (rect3) = 10 cm x 22 cm2 = 220 cm2 x 2 = 440 cm2
Total SA = SA (rect1) + SA (rect2) + SA(rect3)= 240 cm2 + 528 cm2 + 440 cm2
= 1208 cm2
Note Units
B
Example Nets For Triangular Prisms
Surface Area of a triangular prism
15ft
Area Triangle 1 & 2 =
=
=
= 90 ft2
Area Rect. 1 = l x w
= 12 ft x 25 ft
= 300 ft2
Area Rect. 2 & 3 = 25 ft x 20 ft
= 500 ft2
SA = 1480 ft2
SA=triangle1 + triangle2 + rectangle1 + rectangle2 + rectangle3
SA=90 + 90 + 300 + 500+ 500
2 area of same triangle + 2 areas of 3 rectangles, which 2 are the same
Volume of a triangular prism
15ft
Area Triangles =
=
=
= 90 ft2V= Bh
Find area of the base
= (90 ft2)hMultiply it by the height
= (90 ft2) x 25 ft= 2250 ft3
Classwork
• Page 186-188 #4, 5, 6a, 7a• Page 192-193 #4,6,8• Page 198 #4,6a• Page 205 #4, 5
Cylinder
Bases
Cylinder
SA of a Cylinder
Formula for Area of CircleA= r2
= 3.14 x 32
= 3.14 x 9= 28.26
But there are 2 of them so28.26 x 2 = 56.52 units squared
Find the circumference to determine the length of the rectangleC = x d
= 3.14 x 6 (radius doubled)= 18.84
Now use that as your base.A = b x h
= 18.84 x 6 (the height given)= 113.04 units squared
Now add the area of the circles and the area of the rectangle together.
56.52 + 113.04 = 169.56 units squared
The total Surface Area!
Volume of a Cylinder
Formula for Area of CircleA= r2
= 3.14 x 32
= 3.14 x 9= 28.26 unit2
V= Bh
Find area of the base
= (28.26)hMultiply it by the height
= (28.26) x 6= 169.56 unit3
Calculate the SA and Volume of the cylinderBe sure you know the difference between a radius and a diameter!
V = Bh
The radius of the cylinder is 5 m, and the height is 4.2 m; therefore B = A = r2 = 3.14 · 52 = 78.54 m2
V = BhV = 78.54 · 4.2
V = 329.9 m3
Formula for Area of CircleA= r2
= 3.14 x 52
= 3.14 x 25= 78.54
But there are 2 of them so78.54 x 2 = 157.075 units squared
C = x d= 3.1415 x 10 (radius doubled)= 31.15
Now use that as your base.A = b x h
= 31.15 x 4.2 (the height given)= 130.83 units squared
SA = 157.075 + 130.83 = 287.905 m2
Classwork
• Page 211-212 #4,8,9