Transcript PYRAMIDS AND CONES

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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• 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.
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• The other type of pyramid is nonregular.
• In these type of pyramids, you CANNOT draw
a line perpendicular to the base which
intersects the center of the base and the
vertex of the pyramid.
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• We can find the lateral area and surface area
much the same way as we found them in
prisms.
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• The lateral area can 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.
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• 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.
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• Keep in mind that since non-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!
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• The formula for the area of one of the
triangles in a right pyramid is ½ sl with 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 base and l is
the slant height.
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• 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.
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• 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 Pythagorean
theorem to find them.
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• 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.
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• 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.
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• 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.
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• In an oblique cone the perpendicular line
doesn’t pass through the center.
• We won’t be finding the surface area of these
today.
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• The formula for the lateral area of a right cone
is rl where r is the radius of the base l is the
slant height of the cone and r is the radius of
the base.
• That means that the surface area is just
adding in the base or SA = rl + r²
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• The formula for the volume of the cone is just
V = 1/3 Ba where B is the base area and a is
the cone altitude.
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• 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)
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• After this unit, we will learn about cylinders
and you will see that they are very similar in
surface area, lateral area and volume.