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Geometry. Surface Area of Prisms and Cylinders. Goals. Know what a prism is and be able to find the surface area. Know what a cylinder is and be able to find the surface area. Solve problems using prisms and cylinders. Prism. A polyhedron with two congruent faces, called the bases . - PowerPoint PPT Presentation
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Geometry
Surface Area of Prisms and Cylinders
April 19, 2023
Goals
Know what a prism is and be able to find the surface area.
Know what a cylinder is and be able to find the surface area.
Solve problems using prisms and cylinders.
April 19, 2023
Prism
A polyhedron with two congruent faces, called the bases.
The bases are parallel. The other faces are parallelograms and
are called lateral faces. The segments joining corresponding
vertices of the bases are lateral edges.
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ExampleBase
Base
Lateral Face
Lateral Edges
Lateral Face
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Prisms can have any polygon for its bases.
Base is a triangle.
Base is a pentagon.
Triangular Prism
Pentagonal Prism
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These are not prisms:
Lateral Faces are not parallelograms.
…and no parallel bases.
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Altitude of a Prism
The perpendicular distance between the bases.
We usually use the letter h for the height – the length of the altitude.
h
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Right Prism
The lateral edges are perpendicular to the bases.
For clarity, in many cases we do not indicate right prisms – use common sense.
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Oblique Prism
A prism in which lateral faces are not perpendicular to the bases.
110
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Slant Height
The length of a lateral edge in an oblique prism.
Slant HeightsHeight h
Generally, you can use the Pythagorean Theorem to find one or the other.
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Do you know…
What a prism is? What the bases are? What a lateral face is? What the lateral edges are? What a right prism is? What an oblique prism is? What the slant height is?
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Classifying Prisms
Use the shape of the base in the name.
Right Triangular Prism
Right Pentagonal Prism
Right Rectangular Prism
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Have you ever seen a regular heptagonal prism?
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Surface Area
The sum of the areas of all the faces of a prism.
Area = Area of 2 bases + all lateral faces. Contrary to the text, use the symbol SA for
surface area.
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Example6
4
25
The pink sides are really rectangles. They look like parallelograms because of the projection.
There are 2 bases and 4 lateral faces. All are rectangles.
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Example4 46 6
25
6 6
4 4
20
A = 20 25 = 500
? ?
What’s the area?
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Example
Surface Area is the sum of the lateral area (500) and the two bases (48).
4 46 6
25
6 6
4 4
20
A = 20 25 = 500
24 24
SA = 548
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What we did.This measurement is the perimeter of a base. h
P
B BWe found the area of both bases.
We found the rectangular area.
A = Ph
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SA = 2B + Ph
The surface area is the sum of these regions.
h
P
B B
A = Ph
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Surface Area
The surface area of a right prism can be found using
SA = 2B + Ph B is the area of each base P is the perimeter of a base h is the height
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Alternate Method
Find the area of each face separately. Add them together. Don’t omit any face – be careful.
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Lateral Area
The lateral area of a shape is the area of the lateral faces, but doesn’t include the bases.
SA = 2B + Ph is total surface area. Ph is the lateral area. LA = Ph
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Example Find the surface area.
12 ft.
2 ft.
2 ft.
P = 8
h = 12
B = 4
SA = 2(4) + 8(12) = 8 + 96
SA = 104 ft2
or…
2B + Ph
Base
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Example Find the surface area. Alternate solution.
12 ft.
2 ft.
2 ft.
P = 28
h = 2
B = 24
SA = 2(24) + 28(2) = 48 + 56
SA = 104 ft2
2 B + P h
Base
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ExampleAlternate solution 2.
12 ft.
2 ft.
2 ft.
24 ft2
24 ft2
24 ft2
24 ft2 4 ft24 ft2
104 ft2
Separate the figure into a “net”.
Find the area of each face.
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ExampleFind the Surface Area
16
220
B = 40
P = 44
h = 16
SA = 2B + Ph
SA = 2(40) + 44(16)
SA = 80 + 704
S = 784
Base
h
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Your Turn
Find the surface area.
18 cm
7 cm
6 cm
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Solution
18 cm
7 cm
6 cm
Perimeter = 2(6 + 18) = 48 cm
Area = 6 18 = 108 cm2
Base
SA = 2B + Ph = 2(108) + 48(7) = 216 + 336 = 552 cm2
Lateral Area
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Find the Surface Area
6
66
4
236 9 3
4A
2 9 3 18 4
18 3 72
103.2
SA
Area of Equilateral Triangular Base
Surface Area
23
4A s
Hint:
B
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Try this problem.12
10
Find the surface area of the right, hexagonal prism. Each base is a regular polygon.
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Solution12
10
12
?12
?
?
6
12
12 6 3 72
216 3
374.1
A ap
6 3
That’s the area of one base.
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Solution12
10374.1
The perimeter of the hexagon is 6 12 = 72, and the height is 10.
SA = 2B + Ph
SA = 2(374.1) + 72(10)
SA = 748.2 + 720
SA = 1468.3
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Summary
A prism is a polyhedron with 2 congruent bases and parallelogram lateral faces.
Prisms may be right or oblique. Basic Formula: SA = 2B + Ph The Lateral Area LA = Ph
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Cylinders
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Cylinder
A prism with congruent circular bases. May be right or oblique, just like prisms.
r
hr = radius
h = height
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Surface Area of a Cylinder
Take a cylinder and cut it apart…
You get two circles and a rectangular area.
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Surface Area of a Cylinder
The width of the rectangle is…
the circumference of the circle.
h
2r
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Surface Area of a Cylinder
The area of the rectangle is…
2rh
(aka Lateral Area)
h
2r
2rh
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Surface Area of a Cylinder
The area of one circle is…
r2
The area of two circles is 2r2.
h
2r
2rhr
r2
r2
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Surface Area of a Cylinder
The surface area of the cylinder is:
SA = 2r2 + 2rh
h
2r
2rhr
r2
r2
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Surface Area of a Cylinder
h
r22 2
2 ( )
SA r rh
SA r r h
Or, for easier computing…
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Example Find the surface area.
10
12 SA = 2r(r + h)
SA = 2(12)(12 + 10)
SA = 24(22)
SA = 528
SA 1658.76
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Your Turn Find the surface area.
14 in.
d = 2 in.
r = 1 in. SA = 2(1)(1 + 14)
SA = 2(15)
SA = 30
SA 94.25 in2
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Problem. Find the height.
h
4
SA = 301.6
2 ( )
301.6 2 (4)(4 )
301.6 25.13(4 )
301.6 100.52 25.13
201.08 25.13
8
SA r r h
h
h
h
h
h
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Your turn. Find the height.
2 ( )282.74 2 (6)(6 )282.74 12 (6 )7.5 61.5
SA r r hhh
hh
SA = 282.74
6
h
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Take a clean sheet of paper… Label it Chapter 12 Formulas Add these formulas:
Prism Cylinder
SA = 2B + Ph SA=2r(r + h) LA = Ph LA = 2rh Everyday as you have new formulas, add them to
it with a simple drawing.
April 19, 2023
Practice Problems
