FeatureLesson Geometry Lesson Main 1. Draw a net for the figure. Use Euler’s Formula to solve. 2....
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Transcript of FeatureLesson Geometry Lesson Main 1. Draw a net for the figure. Use Euler’s Formula to solve. 2....
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
Notes
11-2
You name a prism by the shape of its bases.
FeatureLesson
GeometryGeometry
LessonMain
Lesson 11-2
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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.
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
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
Surface Areas of Prisms and CylindersSurface Areas of Prisms and Cylinders
Lesson 11-2
Lesson Quiz
11-2