Surface Area and Volume
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SURFACE AREA and VOLUME Solid Figures Introduction CYLINDERS PRISMS PYRAMIDS SPHERES CONES APPENDIX SURFACE AREA and VOLUME SOLID FIGURES In a previous unit we discussed POLYGONS. POLYGONS are 2-dimensional objects. That means they have length and width, but no depth. Like they are cut out of paper. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES In this unit, we will learn about SOLIDS. A SOLID is a 3-dimensional (3D) object. That means it has length, width AND depth. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES When the SOLID is made up of POLYGONS, it is called a POLYHEDRON Each POLYHEDRON is made up of Surfaces (called faces) Segments (called edges) Corners (called vertices) APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES There are many types of solids, we are going to study the 5 basic types: PRISM CYLINDER PYRAMID CONE SPHERE APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES PRISMS: To create a prism, draw any polygon you like. Next, draw another polygon, just like the first one Connect the corresponding vertices BASE BASE BASE APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES PRISMS: APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES PYRAMIDS: Pyramids have one base Instead of connecting to an identical base… They connect to a point. BASE BASE BASE APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES Pyramids: APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES When we name prisms and pyramids, we describe the base to tell what kind of prism or pyramid it is. BASE PRISM PYRAMID Triangle Triangular Prism Triangular Pyramid Rectangle Rectangular Prism Rectangular Pyramid Pentagon Pentagonal Prism Pentagonal Pyramid Hexagon Hexagonal Prism Hexagonal Pyramid Heptagon Heptangonal Prism Heptangonal Pyramid Octagon Octagonal Prism Octagonal Pyramid Decagon Decagonal Prism Decagonal Pyramid APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES CYLINDERS: A cylinder is like a prism… But the bases are circles APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES Cylinders: APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES CONES: A cone is a circular pyramid. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES Cones: APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SPHERES: A sphere is a unique shape... It has no base or sides or edges or vertices. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SPHERES: APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES Identify each of the following shapes as a PRISM, PYRAMID, CYLINDER, CONE or SPHERE #1 #2 #3 SPHERE #5 CYLINDER #4 PRISM #6 PYRAMID #7 CYLINDER #8 PRISM CONE PYRAMID #9 #10 SPHERE CONE & SPHERE APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES So what are we going to do with these shapes? We are going to calculate SURFACE AREA and VOLUME APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES VOLUME: Is the measure of how much “space” an object occupies. It is often thought of as how much water an object can hold. APPENDIX MENU SURFACE AREA and VOLUME SOLID FIGURES VOLUME: Or how many cubic units it would take to fill an object This is why volume is measured in CUBIC UNITS Like: Cubic feet: ft3 Cubic centimeters cm3 APPENDIX MENU SURFACE AREA and VOLUME PRISMS A prism is a solid with 2 congruent BASES connected by LATERAL FACES APPENDIX Late MENU SURFACE AREA and VOLUME PRISMS To find the surface area of a PRISM, you can do it 2 ways: Method 1: Find the area of each shape that makes up the Solid, and add them together 10 6 2 The front 2 x 10 = 20 The back 2 x 10 = 20 The left 6 x 10 = 60 The right 6 x 10 = 60 The top 2x6 = 12 The bottom 2x6 = 12 = 184 APPENDIX MENU SURFACE AREA and VOLUME PRISMS To find the surface area of a PRISM, you can do it 2 ways: Method 2: Use the formula for SURFACE AREA: •Identify the BASEs (in a rectangular prism any pair of opposite sides can be bases) •Find the Area and Perimeter of the base Surface Area • ( S . A.) 2 B ph 10 6 SA 2 12 16 10 24 160 184 2 APPENDIX MENU SURFACE AREA and VOLUME PRISMS Find the surface area of this regular decagonal prism: SA 2 B ph Find the perimeter of the base SA 2 270 60 4 6+6+6+6+6+6+6+6+6+6 =60 Area of the bases Lateral Area 540 240 780m 2 APPENDIX MENU SURFACE AREA and VOLUME PRISMS Finding Volume is easier! Find the VOLUME of this regular decagonal prism: Volume : B h Volume 270 4 1080m 3 APPENDIX MENU SURFACE AREA and VOLUME PRISMS RIGHT PRISMS A right prism is a prism where the lateral faces are all perpendicular to the bases. vs OBLIQUE PRISMS An oblique prism is a prism where the lateral faces are NOT perpendicular to the bases. APPENDIX MENU SURFACE AREA and VOLUME PRISMS SUMMARY: Things you need to know about a prism: A prism has 2 bases All the lateral faces are rectangles Surface Area = 2B + ph B is the area of the base p is the perimeter of the base h is the height of the prism p x h gives the lateral area Volume = B x h B is the area of the base h is the height of the prism APPENDIX MENU SURFACE AREA and VOLUME PRISMS Find the… …Surface Area 3cm 4cm 2cm …Lateral Area …Volume 3cm 4cm 5cm First we have to find that missing side Use the Pythagorean theorem A: 6cm2 P: 12cm APPENDIX MENU SURFACE AREA and VOLUME PRISMS Find the… …Surface Area …Lateral Area 2 B ph 2 6 12 2 12 24 36cm 2 3cm 4cm 5cm 2cm 24cm 2 …Volume Bh 6 2 12cm 3 A: 6cm2 P: 12cm APPENDIX MENU SURFACE AREA and VOLUME PRISMS Find the surface area and volume. SA 2 B ph SA 2 136 52 5 SA 272 260 532 ft 2 V Bh 10 6 24 8 112 14 Area: 136ft2 Perimeter: 52ft V 136 5 680 ft 3 APPENDIX MENU SURFACE AREA and VOLUME PRISMS Find the volume. 9 14 8 1008 1008 880 1888u 2 1110 8 880 APPENDIX MENU SURFACE AREA and VOLUME PRISMS Archimedes problem. APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Since CYLINDERS are like prisms with circular bases, We need to remember a few basic rules for circles. APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the Circumference and area of the circle: Circumference 2 r 2 6 12 37.68 in 6in Area r 2 6 2 36 113.04 in 2 The red segment is the DIAMETER We need the RADIUS APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS A CYLINDER is like a prism with circular bases The surface area of a cylinder can be found with this formula r SA 2B ch Area of the bases Lateral Area h Most people find this formula easier: SA 2 r 2 rh 2 APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the SURFACE AREA of this cylinder: SA 2 r 2 rh 2 2 32 2 3 10 3 ft 18 60 10 ft 78 2 244.92 ft APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS The VOLUME of a Cylinder: The VOLUME of a cylinder can be found with this formula r V Bh h The expanded version of the formula: V r h 2 APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the VOLUME of this cylinder: V r h 2 3 10 9 10 90 2 3 ft 10 ft 282.6ft 3 APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS SUMMARY What you need to know about a cylinder: •A cylinder is a prism with circular bases r •The lateral area is a rectangle •Surface Area: •Volume: SA 2 r 2 rh 2 h V r 2h •Lateral Area: LA 2 rh APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the Lateral Area, Surface Area and Volume: LA 2 rh 2 11 9 198 621.72mm2 9mm SA 2 r 2 rh 2 112 2 11 9 242 198 440 1381.6mm2 2 11mm V r 2h 2 11 9 3419.46mm3 APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the surface area and volume of this shape 31ft 8ft If it was a complete cylinder: SA 2 r 2 rh 2 82 2 8 31 2 1959.36 ft 31ft 2 But we only want half of it 979.68 ft 2 What did we forget? 979.68 496 2 1475.68 ft APPENDIX MENU SURFACE AREA and VOLUME CYLINDERS Find the surface area and volume of this shape 31ft 8ft If it was a complete cylinder: V r h 2 8 31 6,229.76ft 3 2 But we only want half of it 3,114.88ft 3 APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS A PYRAMID is a shape with a single base, where all the vertices of the base are connected to a single point (apex) APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Just like with prisms, we will need to find: height •Base Perimeter •Base Area •Height of the pyramid However, with pyramids, we will also need to find a different measurement… …SLANT HEIGHT APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Slant height is the distance someone would travel walking up the OUTSIDE of a pyramid. APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Finding the SURFACE AREA of a PYRAMID 1 SA B Pl 2 Rectangular Pyramid 25m 22m APPENDIX 22m MENU SURFACE AREA and VOLUME PYRAMIDS Finding the SURFACE AREA of a PYRAMID 1 SA B Pl 2 Rectangular Pyramid 1 484 88 25 2 484 1100 25m Lateral Area 1584m 2 22m 22m Base Area: 484m2 Base Perimeter: 88m APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Finding the VOLUME of a PYRAMID Rectangular Pyramid 11ft 8ft 10ft 80ft2 Base Area: Base Perimeter: 36ft 1 V Bh 3 1 V 80 11 3 293.3ft 3 APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS SUMMARY: Things you need to know about a pyramid: A pyramid has only 1 base All the lateral faces are triangles Surface Area = SA B 1 Pl 2 B is the area of the base p is the perimeter of the base l is the slant height of the pyramid ½ x p x l is the lateral area V h l 1 Bh 3 Volume = B is the area of the base h is the height of the prism APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Hey, I need to know the slant height. 15 Hey, I need to know the height. 12 2 h 2 20 2 2 144 h 400 h 2 256 h 16 82 152 L2 2 289 L 17 L APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Find the Lateral Area, Surface area and Volume: 1 SA B Pl 2 B: P: 400 80 Lateral Area 1 LA Pl 2 1 80 26 1040u 2 2 APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Find the Lateral Area, Surface area and Volume: 1 SA B Pl 2 1 SA 400 80 26 2 B: P: 400 80 SA 1440u 2 APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Find the Lateral Area, Surface area and Volume: 1 V Bh 3 h B: P: 400 80 1 V 400 h 3 102 h 2 262 100 h 2 676 h 576 2 1 V 400 24 3 V 3200u 3 h 24 APPENDIX MENU SURFACE AREA and VOLUME PYRAMIDS Find the Volume: 48 44 1 12 11 3 92u 2 44 4 6 2 48 APPENDIX MENU SURFACE AREA and VOLUME CONES A CONE is basically a pyramid with a circular base APPENDIX MENU SURFACE AREA and VOLUME CONES R: radius H: height L: slant height L H Lateral Area SA B r l or SA r r l 2 R 1 2 V r h 3 APPENDIX MENU SURFACE AREA and VOLUME CONES SA r r l 25 5 13 25 65 2 90 282.6in 2 1 V r2h 3 1 52 12 3 100 Find the slant height with the Pythagorean Theorem 13 12in Find the surface area and volume for the cone shown: 5in 314in 3 APPENDIX MENU SURFACE AREA and VOLUME CONES Find the surface area and volume: SA r r l 92 9 15 678.24u 2 2 1 2 V r h 3 1 9 2 12 3 3052.08u 2 APPENDIX MENU SURFACE AREA and VOLUME CONES Find the Lateral Area: Lateral Area SA B r l 3 5 5 47.1u 2 APPENDIX MENU SURFACE AREA and VOLUME CONES Find the Volume: Cylinder: V 2 r h 2 10 2 18 2 TOTAL: 12,560m3 11304m3 Cone: 1 2 V r h 3 1 10 2 12 3 1256m3 APPENDIX MENU SURFACE AREA and VOLUME SPHERES A sphere is just a ball. R It has 1 important measurement, that is used to find all its other properties The RADIUS R Official definition of a SPHERE: The collection of all points in space that are the same distance (radius) from a point (center). APPENDIX MENU SURFACE AREA and VOLUME SPHERES SA 4 r 2 4 12 2 576 12m 1808.64m 2 4 4 3 V r 123 3 3 2304 7,234.56m APPENDIX 3 MENU SURFACE AREA and VOLUME SPHERES Find the surface area: S 4 r 2 4 (8) 8 in. 2 4 64 804in. APPENDIX 2 MENU SURFACE AREA and VOLUME SPHERES Find the volume: 4 3 V r 3 2 ft. 4 (2)3 3 4 (8) 3 32 3 32 (3.14) 3 34 ft. 3 APPENDIX MENU SURFACE AREA and VOLUME SPHERES Find the volume and surface area of the hemisphere: SA 4 r 2 4 10 400 Half the sphere + the circle 2 200 100 300 942in 2 10in A r2 102 100 APPENDIX MENU SURFACE AREA and VOLUME SPHERES Find the volume and surface area of the hemisphere: SA 4 r 4 V r 3 3 2 10in 4 103 3 4186.7in 3 We only want half 942in 2 2093.3in APPENDIX 3 MENU SURFACE AREA and VOLUME SPHERES Find the volume: 1 2 V r h Cone: 3 1 52 12 3 314mm3 4 4 3 1 3 1 Ice cream: V r 5 2 3 2 3 261.67 mm3 TOTAL: 314 261.67 575.67 APPENDIX MENU SURFACE AREA and VOLUME APPENDIX OPENERS REVIEWS ASSIGNMENTS MENU SURFACE AREA and VOLUME OPENERS A B C D E F G H I J K L M N O P Q R S T U APPENDIX MENU SURFACE AREA and VOLUME OPENERS A APPENDIX MENU SURFACE AREA and VOLUME OPENERS B APPENDIX MENU SURFACE AREA and VOLUME OPENERS C APPENDIX MENU SURFACE AREA and VOLUME OPENERS D APPENDIX MENU SURFACE AREA and VOLUME OPENERS E APPENDIX MENU SURFACE AREA and VOLUME OPENERS F APPENDIX MENU SURFACE AREA and VOLUME OPENERS G APPENDIX MENU SURFACE AREA and VOLUME OPENERS h APPENDIX MENU SURFACE AREA and VOLUME OPENERS I APPENDIX MENU SURFACE AREA and VOLUME OPENERS J APPENDIX MENU SURFACE AREA and VOLUME OPENERS K APPENDIX MENU SURFACE AREA and VOLUME OPENERS L APPENDIX MENU SURFACE AREA and VOLUME OPENERS M APPENDIX MENU SURFACE AREA and VOLUME OPENERS N APPENDIX MENU SURFACE AREA and VOLUME OPENERS O APPENDIX MENU SURFACE AREA and VOLUME OPENERS P APPENDIX MENU SURFACE AREA and VOLUME OPENERS Q APPENDIX MENU SURFACE AREA and VOLUME OPENERS R APPENDIX MENU SURFACE AREA and VOLUME OPENERS S APPENDIX MENU SURFACE AREA and VOLUME OPENERS T APPENDIX MENU SURFACE AREA and VOLUME REVIEWs APPENDIX MENU SURFACE AREA and VOLUME REVIEWs APPENDIX MENU
