# Surface Area of Prisms and Cylinders

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Surface Area of Prisms and Cylinders
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. April 22, 2017

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. April 22, 2017

Example Base Lateral Edges Lateral Face Lateral Face Base
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Prisms can have any polygon for its bases.
Base is a pentagon. Base is a triangle. Pentagonal Prism Triangular Prism April 22, 2017

These are not prisms: …and no parallel bases.
Lateral Faces are not parallelograms. …and no parallel bases. April 22, 2017

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 April 22, 2017

Right Prism The lateral edges are perpendicular to the bases.
For clarity, in many cases we do not indicate right prisms – use common sense. April 22, 2017

Oblique Prism A prism in which lateral faces are not perpendicular to the bases. 110 April 22, 2017

Slant Height The length of a lateral edge in an oblique prism. s
Generally, you can use the Pythagorean Theorem to find one or the other. Slant Height s Height h April 22, 2017

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? April 22, 2017

Classifying Prisms Use the shape of the base in the name.
Right Triangular Prism Right Rectangular Prism Right Pentagonal Prism April 22, 2017

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. April 22, 2017

Example 6 The pink sides are really rectangles. They look like parallelograms because of the projection. 4 25 There are 2 bases and 4 lateral faces. All are rectangles. April 22, 2017

Example 20 What’s the area? A = 20  25 = 500 ? ? 4 6 4 6 25 6 6 4 4
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Example 4 6 4 6 20 Surface Area is the sum of the lateral area (500) and the two bases (48). 25 A = 20  25 = 500 SA = 548 6 6 4 24 4 24 April 22, 2017

What we did. A = Ph P B B This measurement is the perimeter of a base.
We found the rectangular area. We found the area of both bases. B B April 22, 2017

The surface area is the sum of these regions.
P h SA = 2B + Ph A = Ph B B April 22, 2017

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 April 22, 2017

Alternate Method Find the area of each face separately.
Add them together. Don’t omit any face – be careful. April 22, 2017

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 April 22, 2017

Example SA = 104 ft2 2B + Ph Find the surface area. 2 ft. Base 2 ft.
or… April 22, 2017

Example Find the surface area. Alternate solution. Base 2 ft. 2 ft. 12 ft. P = 28 h = 2 B = 24 2 B + P h SA = 2(24) + 28(2) = SA = 104 ft2 April 22, 2017

Example 104 ft2 24 ft2 24 ft2 24 ft2 4 ft2 4 ft2 24 ft2 2 ft.
Alternate solution 2. 2 ft. 12 ft. Separate the figure into a “net”. Find the area of each face. 104 ft2 24 ft2 24 ft2 24 ft2 4 ft2 24 ft2 4 ft2 April 22, 2017

Example Find the Surface Area
B = 40 P = 44 h = 16 SA = 2B + Ph SA = 2(40) + 44(16) SA = S = 784 Base 16 h 2 20 April 22, 2017

Your Turn Find the surface area. 7 cm 6 cm 18 cm April 22, 2017

Solution Perimeter = 2(6 + 18) = 48 cm Base Area = 6  18 = 108 cm2
SA = 2B + Ph = 2(108) + 48(7) = = 552 cm2 Lateral Area April 22, 2017

Find the Surface Area 4 6 B Area of Equilateral Triangular Base
Hint: April 22, 2017

Try this problem. 12 10 Find the surface area of the right, hexagonal prism. Each base is a regular polygon. April 22, 2017

Solution That’s the area of one base. 12 12 12 ? ? ? 6 10
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Solution SA = 2B + Ph SA = 2(374.1) + 72(10) SA = 748.2 + 720
12 374.1 10 The perimeter of the hexagon is 6  12 = 72, and the height is 10. April 22, 2017

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 April 22, 2017

Cylinders April 22, 2017

Cylinder A prism with congruent circular bases.
May be right or oblique, just like prisms. r r = radius h = height h April 22, 2017

Surface Area of a Cylinder
Take a cylinder and cut it apart… You get two circles and a rectangular area. April 22, 2017

Surface Area of a Cylinder
h 2r The width of the rectangle is… the circumference of the circle. April 22, 2017

Surface Area of a Cylinder
2rh h 2r The area of the rectangle is… 2rh (aka Lateral Area) April 22, 2017

Surface Area of a Cylinder
2rh h r2 2r r2 The area of one circle is… r2 The area of two circles is 2r2. April 22, 2017

Surface Area of a Cylinder
2rh h r2 2r r2 The surface area of the cylinder is: SA = 2r2 + 2rh April 22, 2017

Surface Area of a Cylinder
Or, for easier computing… h April 22, 2017

Example Find the surface area.
12 SA = 2r(r + h) SA = 2(12)( ) SA = 24(22) SA = 528 SA  10 April 22, 2017

Your Turn Find the surface area.
d = 2 in. SA = 2(1)(1 + 14) SA = 2(15) SA = 30 SA  in2 r = 1 in. 14 in. April 22, 2017

Problem. Find the height.
4 h SA = 301.6 April 22, 2017

SA = 6 h April 22, 2017

Take a clean sheet of paper…
Label it Chapter 12 Formulas Add these formulas: Prism Cylinder SA = 2B + Ph SA=2r(r + h) LA = Ph LA = 2rh Everyday as you have new formulas, add them to it with a simple drawing. April 22, 2017

Practice Problems April 22, 2017