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Surface Area and Volume

Surface Area and Volume. Chapter 12. 12.1 Exploring Solids. Polyhedron : a solid that is bounded by polygons, called faces , that encolose a single region of space. Edge : the line segment formed by the intersection of two faces. Vertex : a point where 3 or more edges meet.

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Surface Area and Volume

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  1. Surface Area and Volume Chapter 12

  2. 12.1 Exploring Solids • Polyhedron: a solid that is bounded by polygons, called faces, that encolose a single region of space. • Edge: the line segment formed by the intersection of two faces. • Vertex: a point where 3 or more edges meet. • Regular: all faces are congruent regular polygons

  3. Types of Solids • Prism polyhedron • Pyramid polyhedron • Cone not a polyhedron

  4. Types of Solids • Cylinder not a polyhedron • Sphere not a polyhedron

  5. Convex vs. Concave • Convex • Concave (nonconvex)

  6. Cross Sections • If a plane slices through a solid, it forms a cross section. • The cross section of the sphere and the plane is a circle.

  7. Cross Sections • What shape is formed by the intersection of the plane and the solid?

  8. Cross Sections • What shape is formed by the intersection of the plane and the solid?

  9. Platonic Solids • Regular Tetrahedron • 4 faces, 4 vertices, 6 edges • Cube • 6 faces, 8 vertices, 12 edges • Regular Octahedron • 8 faces, 6 vertices, 12 edges • Regular Dodecahedron • 12 faces, 20 vertices, 30 edges • Regular Icosahedron • 20 faces, 12 vertices, 30 edges

  10. Euler’s Theorem • Faces + Vertices = Edges + 2

  11. 12.2 & 12.4 Surface Area and Volume of Prisms and Cylinders • Prism: polyhedron with 2 congruent faces, called bases, that lie in parallel planes. • Lateral faces: parallelograms formed by connecting the bases. • Lateral edges: the segments connecting the vertices of the bases.

  12. More Vocabulary • Right prism: each lateral edge is perpendicular to both bases • Oblique prisms: all prisms that are not right prisms

  13. Surface Area • Find the surface area of a right rectangular prism with a height of 8in., a length of 3in., and a width of 5in.

  14. Surface Area • The surface area (S) of a right prism can be found using the formula S = 2B + Ph, where B is the area of a base, P is the perimeter of a base, and h is the height.

  15. Examples • Find the surface area of the right prism. 6 5 10

  16. Examples • Find the surface area of the right prism. 7 7 5 7

  17. Cylinders • A cylinder is a solid with congruent circular bases that lie in parallel planes. • Lateral area of a cylinder is the area of its curved surface. • Surface area is the sum of the lateral area and the areas of the two bases.

  18. Surface Area of Cylinders • The surface area S of a right cylinder is S = 2B + Ch = 2pr2 + 2prh, where B is the area of the base, C is the circumference of the base, r is the radius of the base, and h is the height.

  19. Examples • Find the surface area of the right cylinder.

  20. Examples • Find the height of a cylinder with a radius of 6.5 and a surface area of 592.19.

  21. Volume of a Solid • Volume of a Cube: V = s3 • Volume Congruence Postulate: • If 2 polyhedra are congruent then their volumes are the same. • Volume Addition Postulate: • The volume of a solid is the sum of the volumes of all its non-overlapping parts.

  22. Volume of a Prism • The volume (V) of a prism is V = Bh, where B is the area of a base and h is the height.

  23. Examples • Find the volume of the prism.

  24. Volume of a Cylinder • The volume V of a cylinder is V = Bh = pr2h, where B is the area of a base, h is the height, and r is the radius of a base.

  25. Examples • Find the volume of the right cylinder.

  26. Examples • Find the volume of the solid.

  27. Examples • Find the volume of the solid.

  28. 12.3 & 12.5 Surface Area & Volume of Pyramids & Cones • Pyramid is a polyhedron with a polygon base and triangular lateral faces with a common vertex. • A regular pyramid has a regular polygon for a base and its height meets the base at its center.

  29. Surface Area of a Regular Pyramid • The surface area S of a regular pyramid is S = B + ½ PL, where B is the area of the base, P is the perimeter of the base, and L is the slant height.

  30. Examples • Find the surface area of the regular pyramid.

  31. Surface Area of a Cone • The surface area S of a right cone is S = pr2 + prL, where r is the radius of the base and L is the slant height.

  32. Examples • Find the surface area of the cone.

  33. Volume of a Pyramid • The volume V of a pyramid is V = 1/3Bh, where B is the area of the base and h is the height.

  34. Example • Find the volume of a pyramid.

  35. Volume of a Cone • The volume V of a cone is V = 1/3 Bh = 1/3pr2h, where B is the area of the base, h is the height, and r is the radius of the base.

  36. Example • Find the volume of the Cone.

  37. Examples • Find the volume of the solid.

  38. Examples • Find the volume of the solid.

  39. 12.6 Volume and Surface area of Spheres • Surface Area of a Sphere: S = 4pr2 • Surface Area = S ; radius = r

  40. Surface Area • Plane intersects a sphere and the intersection contains the center of the sphere, the intersection is a great circle. • Great circles divide the sphere into two hemispheres. • The equator is a great circle.

  41. Using a Great Circle • C = 13.8p ft. for the great circle of a sphere. What is the surface area of the sphere?

  42. 12.6 • Volume of a Sphere: V = 4/3 pr3 • V = volume ; r = radius

  43. Examples

  44. 12.7 Similar Solids • Two solids with equal ratios of corresponding linear measures are called similar solids.

  45. Similar Solids Theorem • If 2 similar solids have a scale factor of a:b, then corresponding areas have a ratio of a2:b2, and corresponding volumes have a ratio of a3:b3.

  46. Examples • The prisms are similar with a scale factor of 1:3. Find the surface area and volume of G if the surface are of F is 24 ft2 and the volume is 7 ft3. G F

  47. Examples • Write the ratio of the two volumes. V = 512 m3 V = 1728 m3

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