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LESSON THIRTY-SIX: DRAW LIKE AN EGYPTIAN

LESSON THIRTY-SIX: DRAW LIKE AN EGYPTIAN. PYRAMIDS AND CONES. So now that we have prisms under our collective belt, we can now begin to understand pyramids. A pyramid is a polyhedron that has a base that can be any polygon and the faces meet at a point called the vertex.

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LESSON THIRTY-SIX: DRAW LIKE AN EGYPTIAN

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  1. LESSON THIRTY-SIX:DRAW LIKE AN EGYPTIAN

  2. PYRAMIDS AND CONES • So now that we have prisms under our collective belt, we can now begin to understand pyramids. • A pyramid is a polyhedron that has a base that can be any polygon and the faces meet at a point called the vertex.

  3. PYRAMIDS AND CONES • As we discussed in the last lesson, pyramids can be slanted or straight. • A straight pyramid is called a regular pyramid. • In these type of pyramids, you can draw a line perpendicular to the base which intersects the center of the base and the vertex of the pyramid.

  4. PYRAMIDS AND CONES • The other type of pyramid is nonregular. • In these type of pyramids, you CANNOTdraw a line perpendicular to the base which intersects the center of the base and the vertex of the pyramid.

  5. PYRAMIDS AND CONES • We can find the lateral area and surface area much the same way as we found them in prisms.

  6. PYRAMIDS AND CONES • The lateral areacan be found by finding the area of all the lateral triangles of the pyramid. • We have to quickly discuss the slant height and altitude of a pyramid.

  7. PYRAMIDS AND CONES • The altitude is line perpendicular to the base which intersects the pyramid’s vertex. • The slant height is a perpendicular bisector to the sides of the base that also intersects the pyramid’s vertex.

  8. PYRAMIDS AND CONES

  9. PYRAMIDS AND CONES • Keep in mind that sincenon-regular pyramids and oblique cones do not have a slant height, we CANNOT use the same formula for the surface area of slanted cones and pyramids. • However, we can find the volume!

  10. PYRAMIDS AND CONES • The formula for the area of one of the triangles in a right pyramid is ½ slwith s equaling the length of a base side and l is the slant height. • So the formula for the total lateral area is ½ Pl where P is the perimeter of the baseand l is the slant height.

  11. PYRAMIDS AND CONES • Therefore, the surface area of the pyramid is just the lateral area plus the base area. • So a workable formula for the surface area of a pyramid is S = ½ Pl + B where B is the area of the base.

  12. PYRAMIDS AND CONES • Keep in mind, that you can find the slant height,altitude and base length given two of the others. • You can use them in the Pythagoreantheorem to find them.

  13. PYRAMIDS AND CONES • The volume of a pyramid can be found by the equation V = 1/3 Ba where B is the area of the base and a is the altitude.

  14. PYRAMIDS AND CONES • You will notice that the formulas for cones are very similar to pyramids. • Since they both come to a vertex, they have very similar qualities.

  15. PYRAMIDS AND CONES • You’ll recall that there are two types of cones. • In regular cones there is a perpendicular line that can be drawn from the center of the circular base though the vertex of the cone.

  16. PYRAMIDS AND CONES • In an oblique cone the perpendicular line doesn’t pass through the center. • We won’t be finding the surface area of these today.

  17. PYRAMIDS AND CONES • The formula for the lateral area of a right cone isrlwhere r is the radius of the base l is the slant height of the coneand r is the radius of the base. • That means that the surface area is just adding in the base or SA = rl + r²

  18. PYRAMIDS AND CONES • The formula for the volume of the cone is just V = 1/3 Ba where B is the base area and ais the cone altitude.

  19. PYRAMIDS AND CONES • As we look back, you can see that all the volume formulas to date are some version of base area times height (altitude). • Prism (V = Bh) • Pyramid (V = 1/3 Ba) • Cone (V = 1/3 Ba)

  20. PYRAMIDS AND CONES • After this unit, we will learn about cylinders and you will see that they are very similar in surface area, lateral area and volume.

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