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LESSON THIRTY-SIX: DRAW LIKE AN EGYPTIANPowerPoint Presentation

LESSON THIRTY-SIX: DRAW LIKE AN EGYPTIAN

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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.

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 CANNOTdraw 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 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 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 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 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 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 Pythagoreantheorem 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 isrlwhere 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 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 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.

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