11.6 Surface Area and Volume of Pyramids and Cones Geometry 11.6 Surface Area and Volume of Pyramids and Cones mbhaub@mpsaz.org

Geometry 12.3 Surface Area of Pyramids and Cones 11.6 Essential Question How do you find the surface area and volume of a pyramid or a cone? November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 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. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones 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.) November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Square Pyramid Vertex Slant Height (l) Altitude (h) November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

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 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Surface Area of a Pyramid Unfold the pyramid: The surface area is the sum of the area of the base and the area of the lateral faces. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Lateral Area Re-arrange the triangles. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Lateral Area ? l S S S S ? Find the area of one triangle. S l November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Lateral Area l S S S S Find the area of four triangles. What is 4s? Perimeter! November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Lateral Area of a Pyramid 𝑃=𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝐵𝑎𝑠𝑒 𝑙=𝑡ℎ𝑒 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Surface Area of a Pyramid Just add the area of the base to the LA. S l November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Volume of a Pyramid Volume of the Prism V = Bh Volume of the Pyramid 𝐵=𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝐵𝑎𝑠𝑒 ℎ=ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Example 1 Name the solid. Then find the SA and V. 12 10 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Finding the slant height. 12 5 10 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Finding the surface area. 10 12 5 13 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Example 2 Name the solid. Then find the SA and V. 14 25 November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Example 3 Name the solid. Then find the SA and V. 12 10 10 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Example 3 Solution ? h 12 5 10 10 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Example 3 Solution Solve for h: 10.91 h 12 5 10 10 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Example 3 Solution Solve for V: 10.91 12 5 10 10 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Example 4 Name the solid. Then find the SA and V. 6 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Example 4 Solution Area of the base: h = 15 6 6 3 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Example 4 Solution Area of the base: h = 15 6 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Circular base No Lateral Faces SA = B + LA Can be right or oblique November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Cone Surface Area You get: Circular Base Cut the cone from the vertex to the base. Then around the base. Circle Sector 1st November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Cone Surface Area A = r2 l Circular Base r 2r The length of this arc is the circumference of the base. l Circle Sector November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Sector Area 2r l Circle Sector This is the lateral area of a cone. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Cone Surface Area A = r2 l Circular Base r 2r A = rl SA = r2 + rl l Circle Sector November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Cone Surface Area SA = r2 + rl LA = rl l h r November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Volume of a Cone Volume of the Cylinder V = r2h Volume of the Cone November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Example 5 Find the SA & V of the right cone. 12 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Example 6 Find theSA & V of the right cone. 15 6 3rd November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Now you can find the surface area and volume of cone or a pyramid! November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones 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. November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Summary - Pyramid h l S These should be on your formula sheet. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Summary - Cone r h l LA = rl SA = r2 + rl Add these to your formula sheet. November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Problem 1 Which figure has the larger volume? 12 14 14 12 November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Problem 2 The base of a square pyramid measures 12 ft on each side, and the height of the pyramid is 15 ft. A cone has a diameter of 12 ft and has the same volume as the square pyramid. What is the height of the cone? November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.5 Volume of Pyramids and Cones Solution Cone Pyramid November 23, 2018 Geometry 12.5 Volume of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Problem 3 Find the volume of the composite figure. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Problem 4 Find the volume of the composite figure. November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Homework November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones

Geometry 12.3 Surface Area of Pyramids and Cones Cone Storage November 23, 2018 Geometry 12.3 Surface Area of Pyramids and Cones