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Mastering Solid Geometry: Surface Area and Volume Study Guide

Enhance your knowledge of solid geometry with comprehensive lessons on calculating surface area and volume of various 3D shapes, including prisms, pyramids, cylinders, cones, and spheres. Practice problems and detailed solutions provided.

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Mastering Solid Geometry: Surface Area and Volume Study Guide

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  1. Materials Needed: Paper, pencil, notes, calculator NO CALCULATOR Surface Area and Volume • By: Mr. Burton • Rockvale High School • Agenda: • Take Notes over Surface Area and Volume • Practice Applying the formulas to 3D objects • Desmos Polygraph Activity • Complete Practice Problems

  2. PART #1 • Surface Area and Volume of a Rectangular Solid • Surface Area and Volume of a Cylinder • Surface Area and Volume of a Cone • Surface Area and Volume of a Sphere

  3. Vocabulary Surface Areais the total area of the surface of a three-dimensional object Volume is the measure of the capacity of a three-dimensional figure. Solid geometry is the study of three­ dimensional solid figures, also called space figures. An edge is a line segment where two faces meet A vertex is a corner point where multiple faces join together A prism is a solid object that has two identical ends and all flat sides. A pyramid is a polyhedron with one base, all of whose faces intersect at a common vertex.

  4. Example 1: Volume and Surface Area • Determine the volume and surface area of the following three-dimensional figure. • Solution

  5. Example 1: Volume and Surface Area • Determine the volume and surface area of the following three­dimensional figure. When appropriate,use the πkey on yourcalculator and roundyour answer to thenearest hundredths.

  6. Example 1: Volume and Surface Area • Solution

  7. Example 1: Volume and Surface Area • Determine the volume and surface area of the following three­dimensional figure. When appropriate,use the πkey on yourcalculator and roundyour answer to thenearest hundredths.

  8. Example 1: Volume and Surface Area • Solution

  9. Example 1: Volume and Surface Area • Determine the volume and surface area of the following three-dimensional figure. When appropriate,use the πkey on yourcalculator and roundyour answer to thenearest hundredths.

  10. Example 1: Volume and Surface Area • Solution

  11. PART #2 • Area of Squares, Rectangles, Circles and Triangles • Surface Area and Volume of a Triangular Prism • Surface Area and Volume of a Pyramid

  12. W A = s² s L s 5 A = 6² = 36 sq. units 12 6 6 Squares and Rectangles Area of Rectangle: A = LW Area of Square: A = s² A = LW Example: Example: A = 12 x 5 = 60 sq. units

  13. r Circles and Triangles Area of Triangle: Area of Circle: A =  r² 9 cm 7 Example: 6 A = (9)² = 81  sq. cm

  14. Formula for Prisms

  15. Volume of a Pyramid • where B is the area of the base and h is the height.

  16. Prism and Pyramids Formulas • Prisms: • Lateral Area: L.A. = ph (p = perimeter, h = height) • Surface Area: S.A. = ph + 2B (B = area of base) • Volume: V = Bh • Regular Pyramids: • Lateral Area: L.A. = ½ lp (p = perimeter, l = slant height) • Surface Area: S.A. = ½ lp + B (B = area of base) • Volume: V = ⅓Bh ( B = area of base, h = height)

  17. 4 4 8 8 4 5 5 6 Volume, SA, and LA in Prisms Examples: h = 8 h = 4 B = 5 x 4 = 20 B = ½ (6)(4) = 12 perimeter of base = 2(5) + 2(4) = 18 perimeter of base = 6 + 5 + 8 = 19 L. A.= 18 x 8 = 144 sq. units L. A. = 19 x 4 = 76 sq. units S.A. = 144 + 2(20) = 184 sq. units S. A. = 76 + 2(12) = 100 sq. units V = 20 x 8 = 160 cubic units V = 12 x 4 = 48 cubic units

  18. vertex Lateral side altitude Base Slant height Regular Pyramids A regular pyramid has a base which is always a regular polygon. • The lateral faces all intersect at a point called the vertex and form triangles. • The altitude is a segment from the vertex perpendicular to the base. • The slant height is the height of a lateral face.

  19. 13 12 10 10 Regular Pyramid( p = perimeter of the base, l = slant height, B = base area) The lateral area of a regular pyramid (L.A.)= ½ lp Surface area of a regular pyramid (S.A.) = ½ lp + B The volume of a right pyramid (V)= ⅓ Bh Example: Perimeter = (2 x 10) + (2 x 10) = 40 Slant height l = 13 ; Height h = 12 Lateral area = ½ (13)(40) = 260 sq. units Area of base = 10 x 10 = 100 sq. units Surface area = 260 + 100 = 360 sq. units Volume = ⅓ (100)(12) = 400 cubic units

  20. 9 10 6 Non-regular Pyramids ** To find the lateral area of a non-regular pyramid, find the area of each face separately. Find the slant height of the front face and the side face. l2= 32 + 92m2 = 52 + 92 l2= 9 + 81 = 90 m2 = 25 + 81 = 106 l = 9.4 m = 10.3 Area of front face = ½ (10)(9.4) = 47 Area of side face = ½ (6)(10.3) = 30.9 m Since the front and back faces are the same, and the two side faces are the same, add the areas of the four faces. l Lateral area = 2(47) + 2(30.9) = 155.8 Continue with the formula for surface area and volume. 3 5

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