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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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pyramids and cones
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.
pyramids and cones1
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.
pyramids and cones2
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.
pyramids and cones3
PYRAMIDS AND CONES
  • We can find the lateral area and surface area much the same way as we found them in prisms.
pyramids and cones4
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.
pyramids and cones5
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.
pyramids and cones7
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!
pyramids and cones8
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.
pyramids and cones9
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.
pyramids and cones10
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.
pyramids and cones11
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.
pyramids and cones12
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.
pyramids and cones13
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.
pyramids and cones14
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.
pyramids and cones15
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
pyramids and cones16
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.
pyramids and cones17
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)
pyramids and cones18
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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