1. Space Figures and Cross Sections
Lesson 11-1
Lesson Quiz
1. Draw a net for the figure.
Sample:
Use Euler’s Formula to solve.
2. A polyhedron with 12 vertices and 30 edges has how many faces? 20
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. 16
Circle
Check students’ drawings; rectangle.
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2. Surface Areas of Prisms and Cylinders
Lesson 11-2
Check Skills You’ll Need
(For help, go to Lessons 1-9 and 10-3.)
Find the area of each net.
Each square has an area of (4)(4) = 16 cm2. The total area is 6(16) = 96 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.
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.
Check Skills You’ll Need
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3. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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.
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4. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
You name a prism by the shape of its bases.
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5. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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 curved
surfaces of a three-dimensional figure. The lateral
area of a prism is the sum of the areas of the lateral faces.
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6. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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 the
perimeter of the base of the prism.
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7. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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!
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8. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
Finding Surface Area of a Prism
Quick Check
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 Area = sum of areas of lateral faces + area of bases
= ( ) + ( )
= 6 • 121
= 726
Because there are six identical faces, the surface area is 726 in.2.
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9. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
Using Formulas to Find Surface Area
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
The triangle has sides of length 18 cm, so p = 3 • 18 cm, or 54 cm.
Draw the base.
Use 30°-60°-90° triangles to find the apothem.
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10. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
Quick Check
(continued)
9 =  a longer leg   shorter leg
9 3 3 9
a =  =
= Rationalize the denominator.
B = ap = 1 2
  54 =
The area of each base of the prism is cm2.
S.A. = L.A B Use the formula for surface area.
= ph + 2B
2( ) Substitute
= (54)(10) + =
Use a calculator.
Rounded to the nearest whole number, the surface area is 821 cm2.
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11. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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.
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12. Surface Areas of Prisms and Cylinders
Lesson 11-2
Notes
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13. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
Finding Surface Area of a Cylinder
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 B Use the formula for surface area of a cylinder.
= rh Substitute the formula for lateral area of a cylinder and area of a circle.
2( r 2)
= (6)(9) (62) Substitute 6 for r and 9 for h.
= Simplify.
= 180 72
The surface area of the cylinder is ft2.
Quick Check
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14. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
Real-World Connection
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 = . d 2
S.A. = L.A B
Cornmeal Container Barley Container
Use the formula for surface area of a cylinder.
= rh r 2
Substitute the formulas for lateral area of a cylinder and area of a circle.
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15. Surface Areas of Prisms and Cylinders
Lesson 11-2
Additional Examples
(continued)
Cornmeal Container Barley Container
S.A. = L.A B
Use the formula for
surface area of a cylinder.
S.A. = L.A B 2
= rh r 2
= rh r 2
Substitute the formulas for lateral area of a cylinder and area of a circle.
= (2)(6) (22 )
= (3)(4) (32 )
Substitute
for r and h. 2 =
Simplify. 8 18
= 32
= 42
Because in.2  in.2, the barley container has the greater surface area.
Quick Check
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17. Surface Areas of Prisms and Cylinders
Lesson 11-2
Lesson Quiz
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. Find
its surface area in terms of .
5. A contractor paints all but the bases of a 28-ft high cylindrical water
tank. The diameter of the base is 22 ft. How many square feet are
painted? Round to the nearest hundred.
S.A. = 216 ft2
S.A. = L.A. + 2B = = 216; 216 ft2
174 cm2
in.2
1900 ft2
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