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.