# SURFACE AREA and VOLUME

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SURFACE AREA and VOLUME
Solid Figures Introduction CYLINDERS PRISMS PYRAMIDS SPHERES CONES APPENDIX

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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

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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

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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

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SOLID FIGURES Before we can do anything with SOLIDS, we must understand the difference between these shapes There are many types of solids, we are going to study the 5 basic types: PRISM PYRAMID CYLINDER CONE SPHERE APPENDIX MENU

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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

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SOLID FIGURES PYRAMIDS: Pyramids have one base Instead of connecting to an identical base… They connect to a point. BASE BASE BASE APPENDIX MENU

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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

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SOLID FIGURES CYLINDERS: A cylinder is like a prism… But the bases are circles APPENDIX MENU

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SOLID FIGURES CONES: A cone is a circular pyramid. APPENDIX MENU

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SOLID FIGURES SPHERES: A sphere is a unique shape... It has no base or sides or edges or vertices. APPENDIX MENU

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SOLID FIGURES Identify each of the following shapes as a PRISM, PYRAMID, CYLINDER, CONE or SPHERE #1 #2 #3 #4 SPHERE CYLINDER PRISM #5 #6 PYRAMID #7 CYLINDER CONE PYRAMID #9 #10 #8 SPHERE CONE & SPHERE PRISM APPENDIX MENU

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SOLID FIGURES So what are we going to do with these shapes? We are going to calculate SURFACE AREA and VOLUME APPENDIX MENU

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SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU

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SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU

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SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU

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SOLID FIGURES SUFARCE AREA: Is literally the area of all the surfaces of a solid. APPENDIX MENU

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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

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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

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PRISMS A prism is a solid with 2 congruent BASES connected by LATERAL FACES Lateral face APPENDIX MENU BASE

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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 The front 2 x 10 = 20 12 The back 2 x 10 = 20 The left 6 x 10 = 60 10 60 The right 6 x 10 = 60 20 The top 2 x 6 = 12 The bottom 2 x 6 = 12 6 = 184 2 APPENDIX MENU

P: perimeter of the base
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 BASE P; 16 A: 12 10 h: height of the prism B: area of the base 6 10 P: perimeter of the base 2 12 16 APPENDIX MENU

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PRISMS Find the surface area of this regular decagonal prism: Find the perimeter of the base =60 Area of the bases Lateral Area A: 270m2 P:60m 6m 4m APPENDIX MENU

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PRISMS Finding Volume is easier! Find the VOLUME of this regular decagonal prism: h: height of the prism B: area of the base A: 270m2 P:60m 6m 4m APPENDIX MENU

RIGHT PRISMS vs OBLIQUE PRISMS
SURFACE AREA and VOLUME PRISMS RIGHT PRISMS vs OBLIQUE PRISMS A right prism is a prism where the lateral faces are all perpendicular to the bases. An oblique prism is a prism where the lateral faces are NOT perpendicular to the bases. APPENDIX MENU

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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 APPENDIX MENU

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PRISMS Find the… …Surface Area 3cm 4cm 2cm …Lateral Area First we have to find that missing side 3cm 4cm …Volume Use the Pythagorean theorem 5cm A: P: 6cm2 12cm APPENDIX MENU

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PRISMS Find the… …Surface Area 3cm 4cm 5cm 2cm …Lateral Area …Volume A: P: 6cm2 12cm APPENDIX MENU

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PRISMS Find the surface area and volume. 10 8 112 24 14 6 Area: Perimeter: 136ft2 52ft APPENDIX MENU

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PRISMS Find the volume. APPENDIX MENU

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CYLINDERS Since CYLINDERS are like prisms with circular bases, We need to remember a few basic rules for circles. APPENDIX MENU

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CYLINDERS Find the Circumference and area of the circle: 6in 12in The red segment is the DIAMETER We need the RADIUS APPENDIX MENU

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CYLINDERS A CYLINDER is like a prism with circular bases The surface area of a cylinder can be found with this formula h: height of the cylinder B: area of the base Area of the bases Lateral Area C: circumference of the base Most people find this formula easier: APPENDIX MENU

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CYLINDERS Find the SURFACE AREA of this cylinder: APPENDIX MENU

h: height of the cylinder
SURFACE AREA and VOLUME CYLINDERS The VOLUME of a Cylinder: The VOLUME of a cylinder can be found with this formula h: height of the cylinder B: area of the base The expanded version of the formula: APPENDIX MENU

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CYLINDERS Find the VOLUME of this cylinder: APPENDIX MENU

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CYLINDERS SUMMARY What you need to know about a cylinder: A cylinder is a prism with circular bases The lateral area is a rectangle Surface Area: Volume: Lateral Area: APPENDIX MENU

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CYLINDERS Find the Lateral Area, Surface Area and Volume: 11mm 9mm APPENDIX MENU

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CYLINDERS Find the surface area and volume of this shape 31ft 8ft 16ft 31ft If it was a complete cylinder: What did we forget? But we only want half of it APPENDIX MENU

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CYLINDERS Find the surface area and volume of this shape 31ft 8ft If it was a complete cylinder: But we only want half of it APPENDIX MENU

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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

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PYRAMIDS Just like with prisms, we will need to find: Base Perimeter Base Area Height of the pyramid However, with pyramids, we will also need to find a different measurement… height Slant height …SLANT HEIGHT APPENDIX MENU

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PYRAMIDS Slant height is the distance someone would travel walking up the OUTSIDE of a pyramid. SLANT HEIGHT APPENDIX MENU

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P: perimeter of the base
SURFACE AREA and VOLUME PYRAMIDS Finding the SURFACE AREA of a PYRAMID Rectangular Pyramid L: Slant height B: area of the base P: perimeter of the base 25m 22m 22m APPENDIX MENU

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PYRAMIDS Finding the SURFACE AREA of a PYRAMID Rectangular Pyramid 25m Lateral Area 22m 22m Base Area: Base Perimeter: 484m2 88m APPENDIX MENU

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PYRAMIDS Finding the VOLUME of a PYRAMID Rectangular Pyramid h: height B: area of the base 11ft 8ft 10ft Base Area: Base Perimeter: 80ft2 36ft APPENDIX MENU

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PYRAMIDS SUMMARY: Things you need to know about a pyramid: A pyramid has only 1 base All the lateral faces are triangles Surface Area = 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 Volume = h is the height of the prism h l APPENDIX MENU

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PYRAMIDS Hey, I need to know the height. Hey, I need to know the slant height. 15 20 24 16 APPENDIX MENU

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PYRAMIDS Find the Lateral Area, Surface area and Volume: Lateral Area B: P: 400 80 APPENDIX MENU

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PYRAMIDS Find the Lateral Area, Surface area and Volume: B: P: 400 80 APPENDIX MENU

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PYRAMIDS Find the Lateral Area, Surface area and Volume: h 10 B: P: 400 80 APPENDIX MENU

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PYRAMIDS Find the Volume: APPENDIX MENU

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CONES A CONE is basically a pyramid with a circular base APPENDIX MENU

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CONES R: radius H: height L: slant height Lateral Area L H R APPENDIX MENU

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CONES Find the surface area and volume for the cone shown: Find the slant height with the Pythagorean Theorem 13 12in 5in APPENDIX MENU

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CONES Find the surface area and volume: APPENDIX MENU

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CONES Find the Lateral Area: Lateral Area 5 APPENDIX MENU

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CONES Find the Volume: TOTAL: Cylinder: Cone: APPENDIX MENU

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SPHERES A sphere is just a ball. It has 1 important measurement, that is used to find all its other properties R 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

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SPHERES Find the surface area: 8 in. APPENDIX MENU

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SPHERES Find the volume: 2 ft. APPENDIX MENU

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SPHERES Find the volume and surface area of the hemisphere: 10in Half the sphere + the circle APPENDIX MENU

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SPHERES Find the volume and surface area of the hemisphere: 10in We only want half APPENDIX MENU

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SPHERES Find the volume: Cone: Ice cream: TOTAL: APPENDIX MENU

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SURFACE AREA and VOLUME REVIEWs