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

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.