1-8 Three-Dimensional Figures Determine whether the solid is a polyhedron. Find the surface area and volume of each solid Then identify the solid. If it is a polyhedron, to the nearest tenth. name the bases, faces, edges, and vertices. 3. 1. SOLUTION: SOLUTION: The formulas for finding the volume and surface A polyhedron is a solid made from flat surfaces that areas of a prism are and , enclose a single region of space. This solid has where S = total surface area, V = volume, h = height curved surfaces, so it is not a polyhedron. The given of a solid, B = area of the base, P = perimeter of the figure is a solid with congruent parallel circular bases base. connected by a curved surface. Therefore, it is a cylinder. Since the base of the prism is a rectangle, the perimeter P of the base is or 14 ANSWER: centimeters. The area of the base B is not a polyhedron; cylinder or 12 square centimeters. The height is 3 centimeters. 2. SOLUTION: The solid is formed by polygonal faces, so it is a polyhedron. It has a rectangular base and three or The surface area of the prism is 66 square more triangular faces that meet at a common vertex. centimeters. So, it is a rectangular pyramid. Base: Faces: Edges: Vertices, K, L, M, N, J The volume of the prism is 36 cubic centimeters. ANSWER: a polyhedron; rectangular pyramid; base: ANSWER: faces edges: 66 cm2 ; 36 cm3 vertices: K, L, M, N, J eSolutions Manual - Powered by Cognero Page 1 1-8 Three-Dimensional Figures 5. CUPCAKES LaMea is icing cupcakes with a cone-shaped icing bag 3.5 inches in diameter, 5 inches tall, with a slant height of approximately 5.3 4. inches. The icing bag has no top. Find each measure SOLUTION: to the nearest tenth. a. the volume of icing that will fill the bag The formulas for finding the volume and surface area b. the area of plastic used to make the icing bag of a sphere are and , where S SOLUTION: = total surface area, V = volume, and r = radius. a. The formula for the volume of a cone is Here, in. where r is the radius of the base and h is the height of the cone. Since the diameter of the cone base is 3.5 inches, the radius is 1.75 inches. The volume of the sphere is or about 904.8 The volume of the icing bag is about 16 cubic inches. cubic inches. b. To find the area of the plastic used to make the icing bag, find the lateral area of the cone. This is the surface area minus the area of the base or , where S = total surface area, V = volume, r = radius, = slant height, and h = height. The surface area of the sphere is or about 452.4 square inches. ANSWER: 144π or about 452.4 in2 ; 288π or about 904.8 in3 The area of the plastic used to make the icing bag is approximately 29.1 square inches. ANSWER: a. ≈16.0 in 3 b. ≈29.1 in2 eSolutions Manual - Powered by Cognero Page 2 1-8 Three-Dimensional Figures Identify the solid modeled by each object. State 9. Refer to Page 80. whether the solid modeled is a polyhedron. SOLUTION: 6. Refer to Page 80. This object models a solid that has two parallel SOLUTION: congruent rectangular bases connected by four This object models a solid with a circular base rectangular faces, so it is a rectangular prism. The connected by a curved surface to a single vertex. So solid is formed by polygonal faces, so it is a it is a cone. A solid with all flat surfaces that enclose polyhedron. a single region of space is called a polyhedron. This solid has a curved surface, so it is not a polyhedron. ANSWER: rectangular prism; a polyhedron ANSWER: cone; not a polyhedron 10. Refer to Page 80. SOLUTION: 7. Refer to Page 80. This object models a solid that is a set of points in space that are the same distance from a given point. SOLUTION: So, it is a sphere. A sphere has no faces, edges, or This object models a solid that has two visible vertices, so it is not a polyhedron. triangular faces that meet at a common vertex. So, it is a pyramid. The type of pyramid will be determined ANSWER: by its base, which is not visible. The base will also sphere; not a polyhedron determine the total number of triangular faces. The solid is formed by polygonal faces, so it is a 11. Refer to Page 80. polyhedron. SOLUTION: ANSWER: This object models a solid with congruent parallel pyramid; a polyhedron circular bases connected by a curved surface. Therefore, it is a cylinder. This solid has a curved 8. Refer to Page 80. surface, so it is not a polyhedron. SOLUTION: ANSWER: This object models a solid has parallel triangular cylinder; not a polyhedron bases connected by three rectangular faces. So, it is a triangular prism. It is formed by polygonal faces, so it is a polyhedron. ANSWER: triangular prism; a polyhedron eSolutions Manual - Powered by Cognero Page 3 1-8 Three-Dimensional Figures STRUCTURE Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices. 13. SOLUTION: A solid with all flat surfaces that enclose a single 12. region of space is called a polyhedron. This solid has a curved surface, so it is not a polyhedron. The given SOLUTION: figure is a solid with a circular base connected by a The solid is formed by polygonal faces, so it is a curved surface to a single vertex. So it is a cone. polyhedron. This solid has two congruent pentagonal bases, so it is a pentagonal prism. ANSWER: Face: Each flat surface is called face. not a polyhedron; cone Edges: The line segments where the faces intersect are called edges. Vertex: The point where three or more edges intersect is called a vertex. Bases: ABCDE, FGHJK Faces: ABCDE, FGHJK 14. SOLUTION: Edges: This solid is formed by polygonal faces, so it is a Vertices: polyhedron. It has triangular bases. So, it is a ANSWER: triangular prism. Face: Each flat surface is called face. a polyhedron; pentagonal prism; bases: ABCDE, Edges: The line segments where the faces intersect FGHJK; faces: ABCDE, FGHJK; are called edges. edges: Vertex: The point where three or more edges intersect is called a vertex. vertices: A, B, C, D, E, F, G, H, J, K Bases: Faces: Edges: Vertices: ANSWER: a polyhedron; triangular prism; bases: faces: edges: ; vertices: M, P, L, J, N, K eSolutions Manual - Powered by Cognero Page 4 1-8 Three-Dimensional Figures 15. 17. SOLUTION: SOLUTION: This solid has no faces, edges, or vertices, so it is not The solid is formed by polygonal faces, so it is a a polyhedron. It is a set of points in space that are the polyhedron. The given pyramid has a pentagonal same distance from a given point. So, it is a sphere. base, so it is a pentagonal pyramid. Faces: Each flat surface is called face. ANSWER: Edges: The line segments where the faces intersect not a polyhedron; sphere are called edges. Vertex: The point where three or more edges intersect is called a vertex. 16. Base: JHGFD Faces: SOLUTION: Edges: A solid with all flat surfaces that enclose a single Vertices: region of space is called a polyhedron. The solid has a curved surface, so it is not a polyhedron. The given ANSWER: figure is a solid with congruent parallel circular bases a polyhedron; pentagonal pyramid; base: JHGFD; connected by a curved surface. Therefore, it is a faces: JHGFD, cylinder. edges: vertices: ANSWER: J, H, G, F, D, E not a polyhedron; cylinder eSolutions Manual - Powered by Cognero Page 5 1-8 Three-Dimensional Figures Find the surface area and volume of each solid to the nearest tenth. 19. SOLUTION: 18. The formulas for finding the volume and surface area SOLUTION: of a prism are and , where S The formulas for finding the volume and surface area = total surface area, V = volume, h = height of a of of a prism are and , where solid, B = area of the base, and P = perimeter of the S = total surface area, V = volume, h = height of a base. solid, B = area of the base, and P = perimeter of the base. Since the base of the prism is a square, the perimeter P of the base is or 18 meters. The area of the Since the base of the prism is a rectangle, the base B is perimeter P of the base is or 14 inches. or 20.25 square meters. The height is 4.5 The area of the base B is or 10 meters. square inches. The height is 6 inches. The surface area of the prism is 121.5 square meters. The surface area of the prism is 104 in2. The volume of the prism is 91.1 cubic meters. The volume of the prism is 60 in3. ANSWER: 121.5 m2 ; 91.1 m3 ANSWER: 104 in2 ; 60 in3 eSolutions Manual - Powered by Cognero Page 6 1-8 Three-Dimensional Figures 20. 21. SOLUTION: SOLUTION: The formulas for finding the volume and surface area The formulas for finding the volume and surface area of a cone are and , of a prism are and , where S = total surface area, V = volume, h = height, B = area where S = total surface area, V = volume, r = radius, of the base, and P = perimeter of the base.