Geometry

Surface Area of Prisms and Cylinders

April 21, 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.

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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.

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Practice Problems