Geometry 11.6 Surface Area and Volume of Pyramids and Cones mbhaub@mpsaz.org
11.6 Essential Question How do you find the surface area and volume of a pyramid or a cone? Geometry 1.3 Surface Area of Pyramids and Cones
Goals Know what a pyramid is. Find the surface area and volume of a pyramid. Know what a cone is. Find the surface area and volume of a cone. Geometry 1.3 Surface Area of Pyramids and Cones 3
Pyramids Geometry 1.3 Surface Area of Pyramids and Cones 4
Pyramid Polyhedron with one polygon base. Lateral faces are triangles. Lateral faces meet at one common vertex. Regular pyramid: base is a regular polygon. (This is what we will discuss.) Geometry 1.3 Surface Area of Pyramids and Cones 5
Square Pyramid Vertex Altitude (h) Slant Height (l) Geometry 1.3 Surface Area of Pyramids and Cones 6
Surface Area of a Pyramid The surface area is the sum of the area of the base and the area of the lateral faces. SA = B + LA Area of Base Area of Lateral Sides Geometry 1.3 Surface Area of Pyramids and Cones 7
Surface Area of a Pyramid S Unfold the pyramid: The surface area is the sum of the area of the base and the area of the lateral faces. Geometry 1.3 Surface Area of Pyramids and Cones 8
Lateral Area Re-arrange the triangles. Geometry 1.3 Surface Area of Pyramids and Cones 9
Lateral Area? l? S S S S Find the area of one triangle. A 1 sl l S Geometry 1.3 Surface Area of Pyramids and Cones 10
Lateral Area l S S S S Find the area of four triangles. A LA 4 1 sl What is 4s? 1 (4 sl ) Perimeter! Geometry 1.3 Surface Area of Pyramids and Cones 11
Lateral Area of a Pyramid LA 1 Pl l P = Perimeter of the Base l = the slant height Geometry 1.3 Surface Area of Pyramids and Cones 1
Surface Area of a Pyramid Just add the area of the base to the LA. l S LA 1 Pl SA B Pl 1 Geometry 1.3 Surface Area of Pyramids and Cones 13
Volume of a Pyramid Volume of the Prism V = Bh Volume of the Pyramid V 1 Bh 3 B = Area of the Base h = height of the pyramid Geometry 1.5 Volume of Pyramids and Cones 14
Example 1 Name the solid. Then find the SA and V. 1 10 Geometry 1.3 Surface Area of Pyramids and Cones 15
Finding the slant height. l 5 1 1 5 144 10 5 169 l 169 13 Geometry 1.3 Surface Area of Pyramids and Cones 16
Finding the surface area. 1 1 13 10 40 13 1 SA B PL 10 5 100 60 360 Geometry 1.3 Surface Area of Pyramids and Cones 17
Example Name the solid. Then find the SA and V. 5 14 Geometry 1.3 Surface Area of Pyramids and Cones 18
Example 3 find the SA and V. Name the solid. Then 1 10 10 Geometry 1.5 Volume of Pyramids and Cones 19
Example 3 Solution? h 1 10 5 10 Geometry 1.5 Volume of Pyramids and Cones 0
Example 3 Solution Solve for h: h 5 1 10.91h 10 5 1 10 h 5 144 h h h 119 119 10.91 Geometry 1.5 Volume of Pyramids and Cones 1
Example 3 Solution 10.91 1 1 10010.91 10 5 Solve for V: 10 V 1 3 3 Bh 363.7 Geometry 1.5 Volume of Pyramids and Cones
Example 4 h = 15 Name the solid. Then find the SA and V. l = 6 7 6 Geometry 1.5 Volume of Pyramids and Cones 3
Example 4 Solution Area of the base: h = 15 6 3 3 6 A 3 1 ap 1 3 3 36 54 3 Geometry 1.5 Volume of Pyramids and Cones 4
Example 4 h = 15 Solution Area of the base: 54 3 6 V 1 1 3 Bh 3 54 3 15 70 3 467.65 Geometry 1.5 Volume of Pyramids and Cones 5
Cones Circular base No Lateral Faces SA = B + LA Can be right or oblique Geometry 1.3 Surface Area of Pyramids and Cones 6
Cone Surface Area You get: Circular Base Cut the cone from the vertex to the base. 1 st Then around the base. Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 7
Cone Surface Area A = r l r r The length of this arc is the circumference of the base. Circular Base l Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 8
Sector Area r A l r l l Circle Sector r l A l A rl This is the lateral area of a cone. Geometry 1.3 Surface Area of Pyramids and Cones 9
Cone Surface Area A = r l r SA = r + rl r Circular Base A = rl l Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 30
Cone Surface Area h r l SA = r + rl LA = rl Geometry 1.3 Surface Area of Pyramids and Cones 31
Volume of a Cone Volume of the Cylinder V = r h Volume of the Cone V 1 3 r h Geometry 1.5 Volume of Pyramids and Cones 3
Example 5 Find the SA & V of the right cone. 1 5 Geometry 1.5 Volume of Pyramids and Cones 33
Example 6 Find thesa & V of the right cone. 15 6 3rd Geometry 1.5 Volume of Pyramids and Cones 34
Now you can find the surface area and volume of cone or a pyramid! Geometry 1.3 Surface Area of Pyramids and Cones 35
Summary The volume of a pyramid is 1/3 the volume of a prism with the same measurements. The volume of a cone is 1/3 the volume of a cylinder with the same measurements. Geometry 1.5 Volume of Pyramids and Cones 36
Summary - Pyramid LA 1 Pl 1 SA B Pl h l V 1 3 Bh S These should be on your formula sheet. Geometry 1.3 Surface Area of Pyramids and Cones 37
Summary - Cone LA = rl SA = r + rl h l V 1 3 r h r Add these to your formula sheet. Geometry 1.5 Volume of Pyramids and Cones 38
Problem 1 Which figure has the larger volume? 1 1 14 14 14 Geometry 1.5 Volume of Pyramids and Cones 39
Problem The base of a square pyramid measures 1 ft on each side, and the height of the pyramid is 15 ft. A cone has a diameter of 1 ft and has the same volume as the square pyramid. What is the height of the cone? Geometry 1.5 Volume of Pyramids and Cones 40
Solution Cone V Pyramid 1 3 1 3 Bh 1 15 14415 70 3 V 1 3 1 3 70 6 70 h 1 70 1 19 r h h h Geometry 1.5 Volume of Pyramids and Cones 41
Problem 3 Find the volume of the composite figure. Geometry 1.3 Surface Area of Pyramids and Cones 4
Problem 4 Find the volume of the composite figure. Geometry 1.3 Surface Area of Pyramids and Cones 43
Homework Geometry 1.3 Surface Area of Pyramids and Cones 44
Cone Storage Geometry 1.3 Surface Area of Pyramids and Cones 45