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.

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 CONES

• 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.

PYRAMIDS AND CONES

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

PYRAMIDS AND CONES

• 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.

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 CONES

PYRAMIDS AND CONES

• 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!

PYRAMIDS AND CONES• 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.

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 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 Pythagorean theorem to find them.

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 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 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 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 CONES• 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

PYRAMIDS AND CONES

• 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.

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 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.