 # Space Figures and Cross Sections

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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. 11-2

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 11-2

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. 11-2

Surface Areas of Prisms and Cylinders
Lesson 11-2 Notes You name a prism by the shape of its bases. 11-2

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. 11-2

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. 11-2

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! 11-2

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. 11-2

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. 11-2

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. 11-2

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. 11-2

Surface Areas of Prisms and Cylinders
Lesson 11-2 Notes 11-2

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 11-2

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. 11-2

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 11-2

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 11-2