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12.3 Surface Area of Pyramids and Cones

12.3 Surface Area of Pyramids and Cones. Advanced Geometry. Pyramid. A pyramid is a polyhedron in which: the base is a polygon the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge .

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12.3 Surface Area of Pyramids and Cones

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  1. 12.3 Surface Area of Pyramids and Cones Advanced Geometry

  2. Pyramid A pyramid is a polyhedron in which: • the base is a polygon • the lateral faces are triangles with a common vertex. • The intersection of two lateral faces is a lateral edge. • The intersection of the base and a lateral face is a base edge. • The altitude, or height, of the pyramid is the perpendicular distance between the base and the vertex.

  3. Pyramid

  4. Regular Pyramid A regular pyramid has a regular polygon for a base and its height meets the base at its center.

  5. Slant Height The slant height of a regular pyramid is the altitude of any lateral face. A nonregular pyramid does not have a slant height.

  6. Example A wooden jewelry box is in the shape of a regular pyramid. Use the diagram of the box to find the area of each lateral face..

  7. Continued What would be the surface area of the regular pyramid in the previous example.

  8. THEOREM 12.4SURFACE AREA OF A REGULAR PYRAMID The surface area S of a regular pyramid is… S = B + (1/2)Pl where B is the area of the base, P is the perimeter of the base, and l is the slant height.

  9. Example Find the surface area of the regular pyramid.

  10. Example A regular pentagonal pyramid has a base with sides of length 18 inches, an apothem of about 12.4 inches, and a slant height of 24 inches. Find the surface area of the pyramid.

  11. CONE A conehas the following: • a circular base • a vertex that is not in the same plane as the base. • The altitude, or height, is the perpendicular distance between the vertex and the base.

  12. CONE

  13. RIGHT CONE In a right cone, the height meets the base at its center and the slant height is the distance between the vertex and a point on the base edge.

  14. Lateral Surface of a Cone

  15. Finding the Lateral Surface Area You can find the area of the sector by using a proportion.

  16. Lateral Area Lateral Area = πrl

  17. THEOREM 12.5SURFACE AREA OF A RIGHT CONE The surface area S of a right cone is S = Area of Base + Lateral Area S = πr² + πrl Where r is the radius of the base and lis the slant height.

  18. Example Find the surface area of the right cone.

  19. Example Find the surface area of a right cone with a radius of 12 feet and a slant height of 16 feet.

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