FeatureLesson Geometry Lesson Main 1. Draw a net for the figure. Use Euler’s Formula to solve. 2. A polyhedron with 12 vertices and 30 edges has how many.

FeatureLesson

GeometryGeometry

LessonMain

1. Draw a net for the figure.

Use Euler’s Formula to solve.2. A polyhedron with 12 vertices and 30 edges has how many faces? 20

16

Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Circle

Check students’ drawings; rectangle.

Sample:

3. A polyhedron with 2 octagonal faces and 8 rectangular faces has how many vertices?

4. Describe the cross section.

5. Draw and describe a cross section formed by a vertical plane cutting the left and back faces of a cube.

Lesson Quiz

11-2

FeatureLesson

GeometryGeometry

LessonMain

(For help, go to Lessons 1-9 and 10-3.)

Find the area of each net.

1. 2. 3.

Lesson 11-2

Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders

Check Skills You’ll Need

Check Skills You’ll Need

Each square has an area of (4)(4) = 16 cm2. The total area is 6(16) = 96 cm2.

The area of each circle is r 2 =(2)2 = 4 cm2. The area of the rectangle is bh = (4)(8) = 32 cm2. The total area of the two circles and the rectangle is 2(4) + 32 = 8 + 32 = 40 , ≈125.7 cm2.

An altitude from any vertex of a triangle measures 3√3. The area of each triangle is ½bh = ½(6)(3√3) = 9√3 m2. The total area of the four triangles is 4(9√3) = 36√3 or about 62 m2.

11-2

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

Prisms and cylinders are 3-D figures which have 2 congruent parallel bases.A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

You name a prism by the shape of its bases.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.

Surface area is the total area of all faces and curvedsurfaces of a three-dimensional figure. The lateralarea of a prism is the sum of the areas of the lateral faces.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to theperimeter of the base of the prism.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces.

Caution!

FeatureLesson

GeometryGeometry

LessonMain

Use a net to find the surface area of the cube.

Draw a net for the cube.

Find the area of one face. 112 = 121

The area of each face is 121 in.2.


Lesson 11-2

Surface Area = sum of areas of lateral faces + area of bases

= (121 + 121 + 121 + 121) + (121 + 121)

= 6 • 121

= 726

Because there are six identical faces, the surface area is 726 in.2.

Quick Check

Additional Examples

Finding Surface Area of a Prism

11-2

FeatureLesson

GeometryGeometry

LessonMain

Find the surface area of a 10-cm high right prism with triangular bases having 18-cm edges. Round to the nearest whole number.

Use the formula L.A. = ph to find the lateral area and the formula S.A. = L.A. + 2B to find the surface area of the prism. The area B of the base is ap, where a is the apothem and p is the perimeter.1

2

Draw the base.

Use 30°-60°-90° triangles to find the apothem.

The triangle has sides of length 18 cm, so p = 3 • 18 cm, or 54 cm.


Lesson 11-2

Additional Examples

11-2

Using Formulas to Find Surface Area

FeatureLesson

GeometryGeometry

LessonMain

9 = 3 a longer leg 3 shorter leg

The area of each base of the prism is 81 3 cm2.


Lesson 11-2

(continued)

S.A. = L.A. + 2B Use the formula for surface area.

= ph + 2B

2(81 3 ) Substitute = (54)(10) +

= 540 + 162 3

820.59223 Use a calculator.

Rounded to the nearest whole number, the surface area is 821 cm2.

Quick Check

81 3B = ap =12

12 3 3 54 =

9 3 3

3 3

9 3

a = = = 3 3 Rationalize the denominator.

Additional Examples

11-2

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-2


Notes

11-2

FeatureLesson

GeometryGeometry

LessonMain

The radius of the base of a cylinder is 6 ft, and its height is 9 ft. Find its surface area in terms of .

S.A. = L.A. + 2B Use the formula for surface area of a cylinder.

= 2 (6)(9) + 2 (62) Substitute 6 for r and 9 for h.

The surface area of the cylinder is 180 ft2.


Lesson 11-2

Quick Check

= 2 rh + Substitute the formula for lateral area of a cylinder and area of a circle.

2( r 2)

= 108 + Simplify.

= 180

72

Additional Examples

11-2

Finding Surface Area of a Cylinder

FeatureLesson

GeometryGeometry

LessonMain

A company sells cornmeal and barley in cylindrical containers. The diameter of the base of the 6-in. high cornmeal container is 4 in. The diameter of the base of the 4-in. high barley container is 6 in. Which container has the greater surface area?

Find the surface area of each container. Remember that r = .d2

S.A. = L.A. + 2B S.A. = L.A. + 2B

Cornmeal Container Barley Container

Use the formula for surface area of a

cylinder.

= 2 rh + 2 r 2 = 2 rh + 2 r 2Substitute the formulas for lateral area of a

cylinder and area of a circle.


Lesson 11-2

Additional Examples

11-2

Real-World Connection

FeatureLesson

GeometryGeometry

LessonMain

S.A. = L.A. + 2B S.A. = L.A. + 2B

Cornmeal Container Barley Container

Use the formula forsurface area of a cylinder.

= 32 = 42

Because 42 in.2 32 in.2, the barley container has the greater surface area.


Lesson 11-2

(continued)

Quick Check

= (2)(6) + (22 ) = (3)(4) + (32 )Substitutefor r and h.

2 2 2 2

= 24 + = 24 + Simplify.8 18

22 = rh + r 2 = rh + r 2Substitute the formulas for lateral area of a cylinder

and area of a circle.

2 2

Additional Examples

11-2

FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

Use the prism below for Exercises 1 and 2.

1.Use a net to find the surface area.

2.Use a formula to find the surface area.

3.The height of a prism is 5 cm. Its rectangular bases have 3-cm and 9-cm sides. Find its surface area.

4.The radius of the base of a cylinder is 16 in., and its height is 4 in.

Findits surface area in terms of .

5.A contractor paints all but the bases of a 28-ft high cylindrical watertank. The diameter of the base is 22 ft. How many square feet arepainted? Round to the nearest hundred.

S.A. = 216 ft2

S.A. = L.A. + 2B = 168 + 48 = 216; 216 ft2

174 cm2

640 in.2

1900 ft2


Lesson 11-2

Lesson Quiz

11-2

FeatureLesson Geometry Lesson Main 1. Draw a net for the figure. Use Euler’s Formula to solve. 2. A polyhedron with 12 vertices and 30 edges has how many.

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